INFSCI 2595 Spring 2023 Homework: 08

Assigned March 21, 2023; Due: March 28, 2023

Collaborators

Include the names of your collaborators here.

Overview

This homework assignment is focused on model complexity and the influence of the prior regularization strength. You will fit non-Bayesian and Bayesian linear models, compare them, and make predictions to visualize the trends. You will use multiple prior strengths to study the impact on the coefficient posteriors and on the posterior predictive distributions.

You are also introduced to non-Bayesian regularization with Lasso regression via the glmnet package. If you do not have glmnet installed please download it before starting the assignment.

IMPORTANT: The RMarkdown assumes you have downloaded the data set (CSV file) to the same directory you saved the template Rmarkdown file. If you do not have the CSV files in the correct location, the data will not be loaded correctly.

IMPORTANT!!!

Certain code chunks are created for you. Each code chunk has eval=FALSE set in the chunk options. You MUST change it to be eval=TRUE in order for the code chunks to be evaluated when rendering the document.

You are free to add more code chunks if you would like.

Load packages

This assignment will use packages from the tidyverse suite as well as the coefplot package. Those packages are imported for you below.

library(tidyverse)

## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.4.0      ✔ purrr   1.0.1 
## ✔ tibble  3.1.8      ✔ dplyr   1.0.10
## ✔ tidyr   1.2.1      ✔ stringr 1.5.0 
## ✔ readr   2.1.3      ✔ forcats 0.5.2 
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()

library(coefplot)

This assignment also uses the splines and MASS packages. Both are installed with base R and so you do not need to download any additional packages to complete the assignment.

The last question in the assignment uses the glmnet package. As stated previously, please download and install glmnet if you do not currently have it.

Problem 01

You will fit and compare 6 models of varying complexity using non-Bayesian methods. The unknown parameters will be be estimated by finding their Maximum Likelihood Estimates (MLE). You are allowed to use the lm() function for this problem.

The data are loaded in the code chunk and a glimpse is shown for you below. There are 2 continuous inputs, x1 and x2, and a continuous response y.

hw_file_path <- 'hw08_data.csv'

df <- readr::read_csv(hw_file_path, col_names = TRUE)

## Rows: 100 Columns: 3
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (3): x1, x2, y
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

df %>% glimpse()

## Rows: 100
## Columns: 3
## $ x1 <dbl> -0.30923281, 0.63127211, -0.68276690, 0.26930562, 0.37252021, 1.296…
## $ x2 <dbl> 0.308779853, -0.547919793, 2.166449412, 1.209703658, 0.785485991, -…
## $ y  <dbl> 0.43636596, 1.37562976, -0.84366730, -0.43080811, 0.77456951, 1.361…

1a)

Create a scatter plot between the response, y, and each input using ggplot().

Based on the visualizations, do you think there are trends between either input and the response?

SOLUTION

p1 <- ggplot(df, aes(x = x1, y = y)) +
  geom_point(size = 2, color = "red") +
  labs(title = "Scatter plot of y vs x1", x = "x1", y = "y")

p2 <- ggplot(df, aes(x = x2, y = y)) +
  geom_point(size = 2, color = "blue") +
  labs(title = "Scatter plot of y vs x2", x = "x2", y = "y")
library(patchwork)
p1 + p2

There is basically no connection between x1 and y, and the points are very scattered. In the x2 diagram, we can clearly see that the closer x2 is to 0, the greater the y value; the closer to both ends, the smaller the y value.

1b)

You will fit multiple models of varying complexity in this problem. You will start with linear additive features which add the effect of one input with the other. Your model therefore controls for both inputs.

Fit a model with linear additive features to predict the response, y. Use the formula interface and the lm() function to fit the model. Assign the result to the mod01 object.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

mod01 <- lm( y ~ x1 + x2, df )
coefplot(mod01)

It can be seen from the figure that the confidence interval of x2 covers both negative and positive values, so it is not credible, and the value of x1 covers all positive values, so it has statistically significant.

1c)

As discussed in lecture, we can derive features from inputs. We have worked with polynomial features and spline-based features in previous assignments. Features can also be derived as the products between different inputs. A feature calculated as the product of multiple inputs is usually referred to as the interaction between those inputs.

In the formula interface, a product of two inputs is denoted by the :. And so if we want to include just the multiplication of x1 and x2 in a model we would type, x1:x2. We can then include main-effect terms by including the additive features within the formula. Thus, the formula for a model with additive features and the interaction between x1 and x2 is:

y ~ x1 + x2 + x1:x2

However, the formula interface provides a short-cut to create main effects and interaction features. In the formula interface, the * operator will generate all main-effects and all interactions for us.

Fit a model with all main-effect and all-interaction features between x1 and x2 using the short-cut * operator within the formula interface. Assign the result to the mod02 object.

Visualize the coefficient summaries with the coefplot() function. How many features are present in the model? Are any of the features statistically significant?

SOLUTION

mod02 <- lm( y ~ x1*x2, df )
coefplot(mod02)

3 features are present in the model. It can be seen from the figure that the confidence intervals of x1, x2, and x1:x2 have not crossed the 0 value, so they all have statistically significant.

1d)

The * operator will interact more than just inputs. We can interact expressions or groups of features together. To interact one group of features by another group of features, we just need to enclose each group within parenthesis, (), and separate them by the * operator. The line of code below shows how this works with the <expression 1> and <expression 2> as place holders for any expression we want to use.

(<expression 1>) * (<expression 2>)

Fit a model which interacts linear and quadratic features from x1 with linear and quadratic features from x2. Assign the result to the mod03 object.

Visualize the coefficient summaries with the coefplot() function. How many features are present in the model? Are any of the features statistically significant?

HINT: Remember to use the I() function when typing polynomials in the formula interface.

SOLUTION

mod03 <- lm( y ~ (x1 + I(x1^2)) * (x2 + I(x2^2)), df )
coefplot(mod03)

As can be seen from the figure, there are eight features in total. Same as above, only the quadratic feature associated with x2 has statistically significant.

1e)

Let’s now try a more complicated model.

Fit a model which interacts linear, quadratic, cubic, and quartic (4th degree) polynomial features from x1 with linear, quadratic, cubic, and quartic (4th degree) polynomial features from x2. Assign the result to the mod04 object.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

mod04 <- lm( y ~ (x1 + I(x1^2) + I(x1^3) + I(x1^4)) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), df )
coefplot(mod04)

Same as above, none of the features has statistically significant.

1f)

Let’s try using spline based features. We will use a high degree-of-freedom natural spline applied to x1 and interact those features with polynomial features derived from x2.

Fit a model which interacts a 12 degree-of-freedom natural (DOF) spline from x1 with linear and quadrtic polyonomial features from x2. Assign the result to mod05.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

mod05 <- lm(y ~ splines::ns(x1, 12) * (x2 + I(x2^2)),df)
coefplot(mod05)

Same as above, none of the features has statistically significant.

1g)

Let’s fit one final model.

Fit a model which interacts a 12 degree-of-freedom natural spline from x1 with linear, quadrtic, cubic, and quartic (4th degree) polyonomial features from x2. Assign the result to mod05.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

mod06 <- lm(y ~ splines::ns(x1, 12) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), df)
coefplot(mod06)

Same as above, none of the features has statistically significant.

1h)

Now that you have fit multiple models of varying complexity, it is time to identify the best performing model.

Identify the best model considering training set only performance metrics. Which model is best according to R-squared? Which model is best according to AIC? Which model is best according to BIC?

HINT: The brooom::glance() function can be helpful here. The broom package is installed with tidyverse and so you should have it already.

SOLUTION

# fit models and extract performance metrics
models <- list(mod01, mod02, mod03, mod04, mod05, mod06)
metrics <- lapply(models, function(mod) {
  broom::glance(mod)
})

# extract performance metrics for each model
r_squared <- sapply(metrics, "[[", "r.squared")
AIC <- sapply(metrics, "[[", "AIC")
BIC <- sapply(metrics, "[[", "BIC")

# print metrics for each model
for (i in seq_along(models)) {
  cat(sprintf("Model %02d:\n", i))
  cat(sprintf("R-squared: %f\n", r_squared[i]))
  cat(sprintf("AIC: %f\n", AIC[i]))
  cat(sprintf("BIC: %f\n", BIC[i]))
  cat("\n")
}

## Model 01:
## R-squared: 0.059447
## AIC: 284.653988
## BIC: 295.074669
## 
## Model 02:
## R-squared: 0.113010
## AIC: 280.790492
## BIC: 293.816342
## 
## Model 03:
## R-squared: 0.546542
## AIC: 223.697344
## BIC: 249.749045
## 
## Model 04:
## R-squared: 0.598728
## AIC: 243.470980
## BIC: 311.205405
## 
## Model 05:
## R-squared: 0.699279
## AIC: 242.625329
## BIC: 346.832136
## 
## Model 06:
## R-squared: 0.781894
## AIC: 262.505077
## BIC: 434.446310

# identify best performing model according to R-squared
best_r_squared <- which.max(r_squared)
cat(sprintf("The best model according to R-squared is Model %02d\n", best_r_squared))

## The best model according to R-squared is Model 06

# identify best performing model according to AIC
best_AIC <- which.min(AIC)
cat(sprintf("The best model according to AIC is Model %02d\n", best_AIC))

## The best model according to AIC is Model 03

# identify best performing model according to BIC
best_BIC <- which.min(BIC)
cat(sprintf("The best model according to BIC is Model %02d\n", best_BIC))

## The best model according to BIC is Model 03

The best model according to R-squared has the highest value, mod06. The best model according to AIC has the lowest value, mod03. The best model according to BIC has the lowest value, mod03.

2a)

You will define a prediction or visualization test grid. This grid will allow you to visualize behavior with respect to x1 for multiple values of x2.

Create a grid of input values where x1 consists of 101 evenly spaced points between -3.2 and 3.2 and x2 is 9 evenly spaced points between -3 and 3. The expand.grid() function is started for you and the data type conversion is provided to force the result to be a tibble.

SOLUTION

viz_grid <- expand.grid(x1 = seq(-3.2, 3.2, length.out = 101),
                        x2 = seq(-3, 3, length.out = 9),
                        KEEP.OUT.ATTRS = FALSE,
                        stringsAsFactors = FALSE) %>% 
  as.data.frame() %>% tibble::as_tibble()

2b)

You will make predictions for each of the models and visualize their trends. A function, tidy_predict(), is created for you which assembles the predicted mean trend, the confidence interval, and the prediction interval into a tibble for you. The result include the input values to streamline making the visualizations.

tidy_predict <- function(mod, xnew)
{
  pred_df <- predict(mod, xnew, interval = "confidence") %>% 
    as.data.frame() %>% tibble::as_tibble() %>% 
    dplyr::select(pred = fit, ci_lwr = lwr, ci_upr = upr) %>% 
    bind_cols(predict(mod, xnew, interval = 'prediction') %>% 
                as.data.frame() %>% tibble::as_tibble() %>% 
                dplyr::select(pred_lwr = lwr, pred_upr = upr))
  
  xnew %>% bind_cols(pred_df)
}

The first argument to the tidy_predict() function is a lm() model object and the second argument is new or test dataframe of inputs. When working with lm() and its predict() method, the functions will create the test design matrix consistent with the training design basis. It does so via the model object’s formula which is contained within the lm() model object. The lm() object therefore takes care of the heavy lifting for us!

Make predictions with each of the six models you fit in Problem 01 using the visualization grid, viz_grid. The predictions should be assigned to the variables pred_lm_01 through pred_lm_06 where the number is consistent with the model number fit previously.

SOLUTION

pred_lm_01 <- tidy_predict(mod01, viz_grid)

pred_lm_02 <- tidy_predict(mod02, viz_grid)

pred_lm_03 <- tidy_predict(mod03, viz_grid)

pred_lm_04 <- tidy_predict(mod04, viz_grid)

pred_lm_05 <- tidy_predict(mod05, viz_grid)

pred_lm_06 <- tidy_predict(mod06, viz_grid)

2c)

You will now visualize the predictive trends and the confidence and prediction intervals for each model. The pred column in of each pred_lm_ objects is the predictive mean trend. The ci_lwr and ci_upr columns are the lower and upper bounds of the confidence interval, respectively. The pred_lwr and pred_upr columns are the lower and upper bounds of the prediction interval, respectively.

You will use ggplot() to visualize the predictions. You will use geom_line() to visualize the mean trend and geom_ribbon() to visualize the uncertainty intervals.

Visualize the predictions of each model on the visualization grid. Pipe the pred_lm_ object to ggplot() and map the x1 variable to the x-aesthetic. Add three geometric object layers. The first and second layers are each geom_ribbon() and the third layer is geom_line(). In the geom_line() layer map the pred variable to the y aesthetic. In the first geom_ribbon() layer, map pred_lwr and pred_upr to the ymin and ymax aesthetics, respectively. Hard code the fill to be orange in the first geom_ribbon() layer (outside the aes() call). In the second geom_ribbon() layer, map ci_lwr and ci_upr to the ymin and ymax aesthetics, respectively. Hard code the fill to be grey in the second geom_ribbon() layer (outside the aes() call). Include facet_wrap() with the facets with controlled by the x2 variable.

To help compare the visualizations across models include a coord_cartesian() layer with the ylim argument set to c(-7,7).

Each model’s prediction visualization should be created in a separate code chunk.

SOLUTION

ggplot(pred_lm_01 ,mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = pred)) +
  facet_wrap(~x2)+
  coord_cartesian(ylim = c(-7, 7))

ggplot(pred_lm_02 ,mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = pred)) +
  facet_wrap(~x2)+
  coord_cartesian(ylim = c(-7, 7))

ggplot(pred_lm_03 ,mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = pred)) +
  facet_wrap(~x2)+
  coord_cartesian(ylim = c(-7, 7))

ggplot(pred_lm_04 ,mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = pred)) +
  facet_wrap(~x2)+
  coord_cartesian(ylim = c(-7, 7))

ggplot(pred_lm_05 ,mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = pred)) +
  facet_wrap(~x2)+
  coord_cartesian(ylim = c(-7, 7))

ggplot(pred_lm_06 ,mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = pred)) +
  facet_wrap(~x2)+
  coord_cartesian(ylim = c(-7, 7))

### 2d)

Do you feel the predictions are consistent with the model performance rankings based on AIC/BIC? What is the defining characteristic of the models considered to be the worst by AIC/BIC?

SOLUTION

What do you think?
I think the predictions doesn’t consistent with the model performance rankings based on AIC/BIC. We know the black line represents the trend, the gray represents the confidence interval, and the orange represents the prediction interval. From model 1 to model 6, we can see that the trend is getting less and less obvious, the confidence interval is getting bigger and bigger, but the prediction interval is shrinking.

AIC and BIC are statistical measures used to compare the goodness of fit of different models. The worst models according to AIC/BIC are those with high AIC/BIC values, indicating that they have poor fit to the data and are too complex relative to the amount of data available.

In simpler terms, AIC/BIC penalizes models that are overly complicated or that do not fit the data well. The more complex a model is, the more likely it is to overfit the data, which means it fits the noise in the data rather than the underlying pattern.

Problem 03

Now that you have fit non-Bayesian linear models with maximum likelihood estimation, it is time to use Bayesian models to understand the influence of the prior on the model behavior.

Regardless of your answers in Problem 02 you will only work with model 3 and model 6 in this problem.

3a)

You will perform the Bayesian analysis using the Laplace Approximation just as you did in the previous assignment. You will define the log-posterior function just as you did in the previous assignment and so before doing so you must create the list of required information. This list will include the observed response, the design matrix, and the prior specification. You will use independent Gaussian priors on the regression parameters with a shared prior mean and shared prior standard deviation. You will use an Exponential prior on the unknown likelihood noise (the \(\sigma\) parameter).

Complete the two code chunks below. In the first, create the design matrix following mod03’s formula, and assign the object to the X03 variable. Complete the info_03_weak list by assigning the response to yobs and the design matrix to design_matrix. Specify the shared prior mean, mu_beta, to be 0, the shared prior standard deviation, tau_beta, as 50, and the rate parameter on the noise, sigma_rate, to be 1.

Complete the second code chunk with the same prior specification. The second code chunk however requires that you create the design matrix associated with mod06’s formula and assign the object to the X06 variable. Assign X06 to the design_matrix field of the info_06_weak list.

SOLUTION

X03 <- model.matrix(y ~ (x1 + I(x1^2)) * (x2 + I(x2^2)), df)

info_03_weak <- list(
  yobs = df$y,
  design_matrix = X03,
  mu_beta = 0,
  tau_beta = 50,
  sigma_rate = 1
)

X06 <- model.matrix(y ~ splines::ns(x1, 12) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), df)

info_06_weak <- list(
  yobs = df$y,
  design_matrix = X06,
  mu_beta = 0,
  tau_beta = 50,
  sigma_rate = 1
)

3b)

You will now define the log-posterior function lm_logpost(). You will continue to use the log-transformation on \(\sigma\), and so you will actually define the log-posterior in terms of the mean trend \(\boldsymbol{\beta}\)-parameters and the unbounded noise parameter, \(\varphi = \log\left[\sigma\right]\).

The comments in the code chunk below tell you what you need to fill in. The unknown parameters to learn are contained within the first input argument, unknowns. You will assume that the unknown \(\boldsymbol{\beta}\)-parameters are listed before the unknown \(\varphi\) parameter in the unknowns vector. You must specify the number of \(\boldsymbol{\beta}\) parameters programmatically to allow scaling up your function to an arbitrary number of unknowns. You will assume that all variables contained in the my_info list (the second argument to lm_logpost()) are the same fields in the info_03_weak list you defined in Problem 3a).

Define the log-posterior function by completing the code chunk below. You must calculate the mean trend, mu, using matrix math between the design matrix and the unknown \(\boldsymbol{\beta}\) column vector.

HINT: This function should look very famaliar…

SOLUTION

lm_logpost <- function(unknowns, my_info)
{
  # specify the number of unknown beta parameters
  length_beta <- ncol(my_info$design_matrix)
  
  # extract the beta parameters from the `unknowns` vector
  beta_v <- unknowns[1:length_beta]
  
  # extract the unbounded noise parameter, varphi
  lik_varphi <- unknowns[length_beta + 1]
  
  # back-transform from varphi to sigma
  lik_sigma <- exp(lik_varphi)
  
  # extract design matrix
  X <- my_info$design_matrix
  
  # calculate the linear predictor
  mu <- as.vector( X %*% as.matrix(beta_v) )
  
  # evaluate the log-likelihood
  log_lik <- sum(dnorm(x = my_info$yobs,
                       mean = mu,
                       sd = lik_sigma,
                       log = TRUE))
  
  # evaluate the log-prior
  log_prior_beta <- sum(dnorm(x = beta_v,
                              mean = my_info$mu_beta,
                              sd = my_info$tau_beta,
                              log = TRUE))
  
  log_prior_sigma <- dexp(x = lik_sigma,
                          rate = my_info$sigma_rate,
                          log = TRUE)
  
  # add the mean trend prior and noise prior together
  log_prior <- log_prior_beta + log_prior_sigma
  
  # account for the transformation
  log_derive_adjust <- lik_varphi
  
  # sum together
  value1<-log_lik + log_prior + log_derive_adjust
  value1
}

3c)

The my_laplace() function is defined for you in the code chunk below. This function executes the laplace approximation and returns the object consisting of the posterior mode, posterior covariance matrix, and the log-evidence.

my_laplace <- function(start_guess, logpost_func, ...)
{
  # code adapted from the `LearnBayes`` function `laplace()`
  fit <- optim(start_guess,
               logpost_func,
               gr = NULL,
               ...,
               method = "BFGS",
               hessian = TRUE,
               control = list(fnscale = -1, maxit = 1001))
  
  mode <- fit$par
  post_var_matrix <- -solve(fit$hessian)
  p <- length(mode)
  int <- p/2 * log(2 * pi) + 0.5 * log(det(post_var_matrix)) + logpost_func(mode, ...)
  # package all of the results into a list
  list(mode = mode,
       var_matrix = post_var_matrix,
       log_evidence = int,
       converge = ifelse(fit$convergence == 0,
                         "YES", 
                         "NO"),
       iter_counts = as.numeric(fit$counts[1]))
}

Execute the Laplace Approximation for the model 3 formulation and the model 6 formulation. Assign the model 3 result to the laplace_03_weak object, and assign the model 6 result to the laplace_06_weak object. Check that the optimization scheme converged.

SOLUTION

laplace_03_weak <- my_laplace(rep(0, ncol(X03)+1), lm_logpost, info_03_weak)
laplace_03_weak

## $mode
##  [1]  0.665297498  0.164476875 -0.160380913 -0.051976413 -0.556454210
##  [6]  0.122777593 -0.082522064  0.005332016  0.020385343 -0.398793318
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  1.382473e-02 -2.857502e-04 -6.974580e-03  3.995613e-04 -5.817298e-03
##  [2,] -2.857502e-04  7.785753e-03  1.697328e-04  8.041568e-04 -5.563936e-04
##  [3,] -6.974580e-03  1.697328e-04  7.581322e-03  1.620125e-04  3.406310e-03
##  [4,]  3.995613e-04  8.041568e-04  1.620125e-04  8.160422e-03  5.829597e-04
##  [5,] -5.817298e-03 -5.563936e-04  3.406310e-03  5.829597e-04  5.670858e-03
##  [6,]  2.435735e-03 -2.044408e-05  1.411797e-04  6.038786e-04 -1.933614e-03
##  [7,] -1.669196e-03 -2.966249e-03  6.048931e-04 -1.077128e-03  1.525096e-03
##  [8,]  1.328275e-03  1.841367e-04 -5.911773e-04 -2.460710e-03 -1.598468e-03
##  [9,]  2.527267e-03  3.147474e-04 -3.492379e-03 -9.078644e-04 -2.395812e-03
## [10,] -8.813906e-09  4.091253e-08 -5.264766e-08  3.626197e-08  8.444137e-08
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,]  2.435735e-03 -1.669196e-03  1.328275e-03  2.527267e-03 -8.813906e-09
##  [2,] -2.044408e-05 -2.966249e-03  1.841367e-04  3.147474e-04  4.091253e-08
##  [3,]  1.411797e-04  6.048931e-04 -5.911773e-04 -3.492379e-03 -5.264766e-08
##  [4,]  6.038786e-04 -1.077128e-03 -2.460710e-03 -9.078644e-04  3.626197e-08
##  [5,] -1.933614e-03  1.525096e-03 -1.598468e-03 -2.395812e-03  8.444137e-08
##  [6,]  1.216999e-02 -4.522317e-03  4.252590e-03 -2.772348e-03 -1.313317e-07
##  [7,] -4.522317e-03  4.979084e-03 -2.548218e-03  8.311808e-04 -1.409092e-08
##  [8,]  4.252590e-03 -2.548218e-03  4.465146e-03 -4.413637e-04 -8.913149e-08
##  [9,] -2.772348e-03  8.311808e-04 -4.413637e-04  3.461386e-03 -1.730025e-08
## [10,] -1.313317e-07 -1.409092e-08 -8.913149e-08 -1.730025e-08  4.999773e-03
## 
## $log_evidence
## [1] -164.7933
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 67

laplace_06_weak <- my_laplace(rep(0, ncol(X06)+1), lm_logpost, info_06_weak)
laplace_06_weak

## $mode
##  [1]  -6.5207834   3.3423300   8.9867547   4.4301616   8.8854915   6.7351266
##  [7]   7.5251127   7.1970111   7.3760587   4.7287927   4.7769825  16.5387116
## [13]   2.3736018   0.5865565   7.8684602  -1.0303226  -3.1185282   0.3019329
## [19]  -3.1150237   2.4434748  -1.9024975   1.1105308  -1.0262699  -1.1203347
## [25]  -2.6808387   0.6611046  -5.3044978   5.1906671  13.0976539   1.0847170
## [31] -17.1669832   3.1155370 -13.0459447  -5.4077670 -10.4130431 -10.9819910
## [37]  -8.0175592  10.2152376 -11.9892792 -23.1224440   0.4810989  -1.5515068
## [43]   2.6250521  -2.5519567   2.4268546  -0.6329237   1.9752677   2.2250425
## [49]  -4.7658272  17.1636200  -5.0155458 -23.6570738 -38.3167356   0.5988171
## [55]   5.3163663   0.4063300   3.9179374   2.7775225   2.5726489   7.3743241
## [61]  -7.0403689  -6.7706288   4.4624827  22.3899530  22.7447404  -0.7577507
## 
## $var_matrix
##                [,1]          [,2]         [,3]          [,4]         [,5]
##  [1,]  13.952023525 -11.100071970 -15.42731051 -12.793671476 -14.64476686
##  [2,] -11.100071970  10.191999074  11.51181182  10.730980494  11.34547555
##  [3,] -15.427310511  11.511811819  17.66158415  13.674469732  16.44980855
##  [4,] -12.793671476  10.730980494  13.67446973  12.546784353  12.98690314
##  [5,] -14.644766864  11.345475554  16.44980855  12.986903143  15.82076980
##  [6,] -13.585726001  10.991299618  14.86246020  12.788665856  13.91758005
##  [7,] -14.168865584  11.146368668  15.77859636  12.761046161  15.09584219
##  [8,] -13.862804719  11.090298473  15.27565337  12.820168943  14.44177219
##  [9,] -14.030905448  11.131504607  15.54228907  12.809663349  14.78206988
## [10,] -13.774947743  10.995569290  15.20193130  12.671585604  14.42478163
## [11,]  -7.313207088   6.218197212   7.86540220   6.852250971   7.60652331
## [12,] -30.174218176  22.998573806  33.93075302  27.274033902  31.87973501
## [13,]  -3.974803841   3.789010650   4.04251707   3.901970686   4.01007045
## [14,] -17.615596135  17.055646467  17.69342268  16.898695451  18.07847003
## [15,]  -5.446445631   1.272459352   7.68034937   4.201892441   6.15206640
## [16,]   9.909347166  -9.846709771  -9.77314875  -9.635065995 -10.10646735
## [17,]  -0.760357646   2.008580493   0.11417643   1.014224941   0.61865817
## [18,]  17.062201666 -16.640288767 -17.06249770 -16.478484534 -17.47520120
## [19,]  18.063053518 -17.163620021 -18.43800514 -16.894783675 -18.67225376
## [20,]  16.900376852 -16.935373508 -16.45467248 -17.287426906 -16.94148139
## [21,]  17.803314770 -17.044642969 -18.06391259 -16.689626285 -18.41958706
## [22,]  17.514141201 -17.121253046 -17.42503081 -17.233894306 -17.70430506
## [23,]  18.023613431 -17.131987524 -18.36993574 -16.832465310 -18.81911933
## [24,]  17.611233328 -17.045045827 -17.69105478 -16.891836878 -18.11768202
## [25,]  17.580834822 -17.064126218 -17.62546414 -16.925701966 -17.96155655
## [26,]  17.661788147 -17.032993359 -17.78361907 -16.909562551 -18.17838954
## [27,]  10.553316210  -9.977748804 -10.74542225 -10.027585780 -10.86508522
## [28,]  34.931539638 -34.386321132 -34.75744987 -33.700374435 -35.74475091
## [29,]   5.636228740  -5.495633914  -5.57122380  -5.581851416  -5.73948627
## [30,]   0.060843800  -0.848096827   0.36859704  -0.992421705   0.44122207
## [31,]   8.034716837  -1.016028350 -12.47647017  -4.590978893  -9.80347932
## [32,]   2.164668707  -0.646282515  -2.29441716  -4.351439741  -1.27075065
## [33,]   7.204258377  -1.768392850 -10.40797502  -4.426896734  -9.11763426
## [34,]   4.931595281  -1.279583946  -6.70572910  -4.757804052  -4.73224151
## [35,]   5.563769227  -1.163388628  -8.02276622  -3.677553480  -6.80523620
## [36,]   5.359732430  -1.393052103  -7.40930013  -4.527727692  -5.70727983
## [37,]   5.543082611  -1.242798191  -7.89532757  -4.019390264  -6.48015276
## [38,]   4.363227292  -0.582645448  -6.32063716  -3.447528750  -4.88921614
## [39,]   1.869597900  -0.971387570  -2.37490859  -1.710109797  -1.96746077
## [40,]  14.397628409  -2.089338146 -20.98970934 -10.291842683 -16.67132173
## [41,]  -2.871925691   1.499579480   3.63382737   1.993711673   3.41278910
## [42,]  -7.615774168   8.627972507   6.99019434   7.821875043   7.53596950
## [43,] -10.814243986  10.349967628  10.87853353  10.229020981  11.15418057
## [44,]  -8.752978088   9.508262023   7.98912305   9.482269885   8.49766975
## [45,] -10.431792541  10.003006379  10.57371519   9.738729826  10.86219183
## [46,]  -9.703240201   9.851933770   9.38338898   9.840229917   9.59381004
## [47,] -10.344041265   9.943968854  10.47093537   9.630485203  10.87829920
## [48,] -10.018976568   9.856687034   9.96560042   9.582264479  10.37292986
## [49,]  -9.387242931   9.591909626   9.03982573   9.477497103   9.28934986
## [50,] -10.734554703  10.301031376  10.85881561  10.170483670  11.12118330
## [51,]  -4.468150931   4.384610681   4.41690719   4.379748959   4.51626077
## [52,] -19.658192992  20.749781925  18.58448381  19.752455033  19.72043001
## [53,]   4.372884416  -2.984250336  -5.31552872  -3.298744407  -4.91620471
## [54,]   1.535423525  -1.452910157  -1.59815593  -1.226075491  -1.70195805
## [55,]   0.362426805  -2.304631885   0.83012982  -1.199438822   0.04752874
## [56,]   1.418162586  -2.046590874  -1.28680037  -0.691658347  -1.68421831
## [57,]   0.471947481  -1.964636900   0.37502225  -1.121071618  -0.04159788
## [58,]   0.497316621  -1.812370816   0.15608641  -0.503073438  -0.57823077
## [59,]   1.161796567  -2.238507555  -0.60895831  -1.487896734  -0.94300406
## [60,]   0.546053548  -1.774121487   0.06451224  -0.653356343  -0.55825062
## [61,]   1.401289228  -2.520419377  -0.82381599  -1.666557529  -1.18153546
## [62,]   1.212602611  -2.254587118  -0.71282227  -1.299923097  -1.17491896
## [63,]   0.319914612  -0.419470620  -0.27223852  -0.256600385  -0.34106495
## [64,]   0.298098338  -4.847982014   2.15128985  -1.725308394   0.47290254
## [65,]  -2.814844801   2.528698939   3.04188180   2.497026481   2.95916485
## [66,]   0.008362952  -0.003691296  -0.01102439  -0.006423542  -0.00944052
##                [,6]          [,7]         [,8]          [,9]        [,10]
##  [1,] -13.585726001 -14.168865584 -13.86280472 -14.030905448 -13.77494774
##  [2,]  10.991299618  11.146368668  11.09029847  11.131504607  10.99556929
##  [3,]  14.862460201  15.778596363  15.27565337  15.542289066  15.20193130
##  [4,]  12.788665856  12.761046161  12.82016894  12.809663349  12.67158560
##  [5,]  13.917580047  15.095842188  14.44177219  14.782069879  14.42478163
##  [6,]  13.775049903  13.402205037  13.72234630  13.549238277  13.47446204
##  [7,]  13.402205037  15.291453973  13.39887785  14.604481414  13.80280589
##  [8,]  13.722346301  13.398877851  14.64531691  13.469969537  13.92128767
##  [9,]  13.549238277  14.604481414  13.46996954  14.738821918  13.39170517
## [10,]  13.474462042  13.802805890  13.92128767  13.391705169  14.89338252
## [11,]   7.136074173   7.515326692   7.11561964   7.689680419   6.36975252
## [12,]  29.274678961  30.645206790  30.06331337  30.123032167  30.25350429
## [13,]   4.024850709   3.733480535   4.31697573   3.324646765   5.37213896
## [14,]  17.504492114  17.641682105  17.67198946  17.672228449  17.37718612
## [15,]   4.951776497   5.777268105   5.25292932   5.526915831   5.35286258
## [16,]  -9.887540123  -9.890787396  -9.95780129  -9.933871841  -9.77922883
## [17,]   0.893033404   0.668828355   0.82538395   0.744168111   0.76096423
## [18,] -16.980973209 -17.062719287 -17.12308339 -17.108300763 -16.84652180
## [19,] -17.832013290 -18.210241432 -18.07433576 -18.153082540 -17.78700563
## [20,] -17.112938077 -16.638582788 -17.07199770 -16.881820801 -16.73827767
## [21,] -17.550214353 -17.974048985 -17.80769332 -17.896712072 -17.53293533
## [22,] -17.695573658 -17.273005459 -17.60605072 -17.571330333 -17.19054894
## [23,] -17.605847314 -18.319284295 -18.05417993 -18.078440938 -17.90147620
## [24,] -17.286634373 -18.619916370 -16.33725125 -18.462708209 -16.92848245
## [25,] -17.748465404 -16.440113380 -19.16443924 -16.643883345 -18.18149725
## [26,] -17.357855091 -18.539120821 -16.62418254 -18.491761814 -15.20933589
## [27,] -10.561538951 -10.044115868 -11.32038341 -10.258619958 -10.97341670
## [28,] -34.743772554 -35.168399471 -34.72374011 -35.036542486 -35.21641399
## [29,]  -5.492536272  -6.401230617  -4.53689579  -5.882530818  -6.46356735
## [30,]  -0.248511074   0.086034082  -0.08678867  -0.011087859  -0.17025302
## [31,]  -6.870155132  -8.902277401  -7.60753376  -8.256082183  -7.76274320
## [32,]  -2.942039385  -1.498826006  -2.40867410  -1.997451792  -2.27790963
## [33,]  -5.754314481  -8.242518184  -6.69371365  -7.457099298  -6.97693326
## [34,]  -5.721730111  -4.291922772  -5.18528365  -4.791602793  -4.91201574
## [35,]  -4.196455699  -7.350850015  -4.33838798  -6.189075443  -5.24951931
## [36,]  -5.519432032  -3.931332040  -7.45481633  -4.205634110  -5.33771465
## [37,]  -4.580615005  -7.303722037  -3.42085953  -7.988469440  -4.88327389
## [38,]  -4.001863519  -4.589196046  -4.18314859  -3.463293773  -9.06232012
## [39,]  -1.759632404  -1.875977004  -1.89143480  -2.497555199   0.74904233
## [40,] -12.972432066 -15.165030433 -14.19993422 -13.599142561 -16.30587247
## [41,]   2.483505642   3.442778444   2.24266129   5.154444444  -2.86598201
## [42,]   7.730113898   7.509047744   7.69702376   7.611195427   7.54945222
## [43,]  10.703403141  10.865701326  10.83717691  10.857822762  10.65533607
## [44,]   8.993601348   8.514641536   8.88481913   8.722099157   8.70254305
## [45,]  10.272885212  10.529351827  10.44416553  10.478794166  10.27192386
## [46,]   9.974975619   9.462673695   9.74556772   9.771030368   9.43859381
## [47,]  10.046678872  10.451442528  10.50659963  10.257279106  10.40869949
## [48,]   9.585866449  11.541778526   8.05316218  11.303350307   9.19991951
## [49,]  10.042289218   6.917202449  12.53849870   6.966960124  11.45790957
## [50,]  10.383325486  11.884212300   9.44786014  11.900578520   5.27823975
## [51,]   4.640311526   3.583303271   5.56685496   4.001034683   6.72955764
## [52,]  19.777828417  20.012648416  18.96861661  18.437786848  25.63081767
## [53,]  -4.285931613  -3.034535033  -6.52951777  -6.202327205   4.66937778
## [54,]  -1.484463653  -1.574396291  -1.53790802  -1.552794035  -1.48936071
## [55,]  -0.683813279  -0.118029472  -0.49442410  -0.308366484  -0.40212028
## [56,]  -1.150518692  -1.644276770  -1.34476572  -1.474845876  -1.38510429
## [57,]  -0.882016061  -0.170969082  -0.62676673  -0.411505341  -0.48663010
## [58,]  -0.212028382  -0.719696924  -0.45588965  -0.509407869  -0.57661444
## [59,]  -1.552513220  -0.635402585  -1.54453455  -1.004302087  -1.08226578
## [60,]  -0.307801307  -1.650641954   0.96869667  -1.303454470  -1.01300300
## [61,]  -1.900867552   0.200249724  -3.52525246   0.003007229  -0.03916132
## [62,]  -1.186052187  -1.637684163  -0.69962166  -1.825943322   1.78667677
## [63,]  -0.402568011   0.065721902  -0.81266430  -0.030800667  -1.61302786
## [64,]  -0.815454287  -0.250379214   0.01083985   0.185097243  -3.18508068
## [65,]   2.866307410   2.003558401   4.06334571   3.306508599  -1.11195255
## [66,]  -0.007718723  -0.008465339  -0.00874500  -0.009037369  -0.00486937
##               [,11]         [,12]         [,13]         [,14]       [,15]
##  [1,]  -7.313207088 -30.174218176  -3.974803841  -17.61559614  -5.4464456
##  [2,]   6.218197212  22.998573806   3.789010650   17.05564647   1.2724594
##  [3,]   7.865402197  33.930753024   4.042517069   17.69342268   7.6803494
##  [4,]   6.852250971  27.274033902   3.901970686   16.89869545   4.2018924
##  [5,]   7.606523309  31.879735005   4.010070450   18.07847003   6.1520664
##  [6,]   7.136074173  29.274678961   4.024850709   17.50449211   4.9517765
##  [7,]   7.515326692  30.645206790   3.733480535   17.64168210   5.7772681
##  [8,]   7.115619638  30.063313368   4.316975726   17.67198946   5.2529293
##  [9,]   7.689680419  30.123032167   3.324646765   17.67222845   5.5269158
## [10,]   6.369752525  30.253504287   5.372138964   17.37718612   5.3528626
## [11,]   5.121309926  14.394038662  -0.100695563   10.55944150   1.7010489
## [12,]  14.394038662  67.935654304  11.312037674   34.81134234  14.6764897
## [13,]  -0.100695563  11.312037674   9.029255245    7.11261362  -0.4484109
## [14,]  10.559441499  34.811342342   7.112613618   64.85442118 -27.7611279
## [15,]   1.701048871  14.676489674  -0.448410879  -27.76112786  37.9286779
## [16,]  -6.105078647 -19.177513427  -4.218990758  -38.37243595  16.2347846
## [17,]   0.873526226   0.444241164   1.092684429   17.38655308 -17.8226954
## [18,] -10.237869928 -33.685761035  -6.918870937  -59.76746358  25.8312545
## [19,] -10.731529658 -35.948611804  -7.179126200  -66.85714696  28.5700146
## [20,] -10.307142820 -32.921053865  -7.054649841  -63.33134958  27.7040955
## [21,] -10.605441880 -35.358441017  -7.141036700  -65.30902736  27.8054156
## [22,] -10.601638528 -34.444627133  -7.060924703  -64.78701270  27.9108573
## [23,] -10.619776437 -35.935579438  -7.278560024  -65.23691699  27.4065078
## [24,] -10.749232186 -34.741556882  -6.855069628  -64.69574415  27.6451219
## [25,] -10.176969484 -34.860754934  -7.647543854  -65.00414181  27.9379705
## [26,] -11.778315627 -34.443656816  -5.356285443  -64.49561408  27.3658165
## [27,]  -5.083014189 -21.885303756  -6.253987318  -35.90229819  14.3485243
## [28,] -21.157626537 -68.420486480 -14.444248317 -136.10893269  60.9067386
## [29,]  -4.137042449 -10.586859782  -0.672665658  -17.84116589   6.7818590
## [30,]  -0.070408125  -0.002670521   0.070414230   25.62031802 -24.5470106
## [31,]  -2.341575908 -22.138748476   0.808483385   30.65528907 -44.6720887
## [32,]  -0.538482071  -6.080005535   0.513058450   29.83931898 -34.2496419
## [33,]  -2.396746992 -19.133105964   0.365214361   26.35911590 -39.5786817
## [34,]  -1.521708760 -13.255033916   0.342475104   27.54819947 -36.8608269
## [35,]  -1.846727809 -14.996284574   0.792736459   28.18118426 -38.5219507
## [36,]  -1.545437700 -14.528844336  -0.069101549   27.27614210 -37.3659253
## [37,]  -2.203703368 -14.709916487   1.248161597   27.74011901 -38.0464713
## [38,]   1.010145041 -12.169680475  -0.702054674   30.17505018 -38.2443581
## [39,]  -3.405316281  -2.814494890   4.042749140   13.10173031 -16.6982062
## [40,]   0.411762089 -47.842678882 -13.307209684   62.15415284 -88.7573107
## [41,]  10.068105802  -6.333590093 -28.986649463   12.55307799  -6.9996032
## [42,]   5.014609818  13.921412627   3.772955046   31.85833668 -16.0767321
## [43,]   6.541099160  21.238443317   4.417969276   41.07456033 -16.6622891
## [44,]   5.653487685  16.252647406   4.116682277   36.74848663 -16.8177166
## [45,]   6.306293142  20.499552419   4.273640481   39.07063768 -15.9548683
## [46,]   6.097732384  18.597777274   4.196542917   38.34638496 -16.5886084
## [47,]   6.184556585  20.310123809   4.303925136   38.67197575 -15.7633975
## [48,]   6.352243608  19.505001434   4.120583140   38.37085097 -16.0734480
## [49,]   5.234501891  17.761771844   4.273011862   37.42437386 -16.2335361
## [50,]   8.440539264  21.577839477   3.835460721   39.82322697 -16.1506969
## [51,]   1.402540329   8.181551461   1.914216914   15.62266077  -5.4671268
## [52,]   7.399377066  44.264339084  25.161306738   86.49073876 -42.0093453
## [53,] -10.018446476   3.088368689  26.260980744  -10.00434450   1.8975473
## [54,]  -0.896648322  -3.090650723  -0.686695934  -14.33297741  12.4856343
## [55,]  -0.863264553   0.928888714  -1.284629639  -18.95233797  20.0048019
## [56,]  -1.073848978  -2.242441260  -1.066917908  -17.65896008  16.8927968
## [57,]  -0.774076907   0.318658138  -1.096621177  -17.24354182  18.1926227
## [58,]  -0.715395931   0.124572804  -1.010806578  -16.69881366  17.6138523
## [59,]  -1.062019971  -1.396538510  -1.239347483  -18.14878073  17.8606101
## [60,]  -0.650305650   0.053900370  -0.885425865  -16.53206037  17.4380732
## [61,]  -1.504353976  -1.992369248  -1.426641588  -19.15954813  18.2864862
## [62,]  -2.264499205  -1.824272918  -0.909378614  -18.20703539  17.8000529
## [63,]   0.780978343  -0.616090353  -0.713770334   -6.14281601   6.3959035
## [64,]  -0.105162870   1.576928492  -7.538508677  -41.40845344  45.2508728
## [65,]   3.962686619   2.117420499  -6.265636012    6.26894689  -0.1324686
## [66,]  -0.005279742  -0.018512983   0.004822222    0.01513112  -0.0293585
##              [,16]        [,17]        [,18]        [,19]        [,20]
##  [1,]   9.90934717  -0.76035765  17.06220167  18.06305352  16.90037685
##  [2,]  -9.84670977   2.00858049 -16.64028877 -17.16362002 -16.93537351
##  [3,]  -9.77314875   0.11417643 -17.06249770 -18.43800514 -16.45467248
##  [4,]  -9.63506599   1.01422494 -16.47848453 -16.89478367 -17.28742691
##  [5,] -10.10646735   0.61865817 -17.47520120 -18.67225376 -16.94148139
##  [6,]  -9.88754012   0.89303340 -16.98097321 -17.83201329 -17.11293808
##  [7,]  -9.89078740   0.66882835 -17.06271929 -18.21024143 -16.63858279
##  [8,]  -9.95780129   0.82538395 -17.12308339 -18.07433576 -17.07199770
##  [9,]  -9.93387184   0.74416811 -17.10830076 -18.15308254 -16.88182080
## [10,]  -9.77922883   0.76096423 -16.84652180 -17.78700563 -16.73827767
## [11,]  -6.10507865   0.87352623 -10.23786993 -10.73152966 -10.30714282
## [12,] -19.17751343   0.44424116 -33.68576103 -35.94861180 -32.92105386
## [13,]  -4.21899076   1.09268443  -6.91887094  -7.17912620  -7.05464984
## [14,] -38.37243595  17.38655308 -59.76746358 -66.85714696 -63.33134958
## [15,]  16.23478456 -17.82269536  25.83125451  28.57001461  27.70409550
## [16,]  23.18907834  -9.69819839  35.21187906  39.41329313  37.69047521
## [17,]  -9.69819839   9.52735581 -16.50328430 -17.97508600 -17.05031496
## [18,]  35.21187906 -16.50328430  56.53648888  60.89992118  58.86415185
## [19,]  39.41329313 -17.97508600  60.89992118  70.56789120  63.53981596
## [20,]  37.69047521 -17.05031496  58.86415185  63.53981596  65.35689152
## [21,]  38.55257779 -17.51125849  60.00313815  68.02507052  62.37332379
## [22,]  38.40367666 -17.38883102  59.85975974  66.18488315  64.47515598
## [23,]  38.50589157 -17.36631495  60.01712302  67.73095492  62.70139277
## [24,]  38.28509491 -17.32445911  59.64072372  66.60934762  63.32422798
## [25,]  38.46668473 -17.46148721  59.90455304  67.01823588  63.48447417
## [26,]  38.15438678 -17.20615019  59.44637229  66.45346400  63.02479044
## [27,]  21.18607324  -9.34590620  33.40729733  36.98409016  34.87314432
## [28,]  80.64405249 -37.24881551 124.62326971 140.53728633 133.07017415
## [29,]  10.58849868  -4.52024050  17.00552532  17.77797200  18.27351545
## [30,] -14.39099057  12.68778717 -23.01850979 -27.42060806 -23.60246231
## [31,] -18.70670720  20.03422469 -28.94351257 -29.04061721 -34.32634244
## [32,] -16.76288293  17.22241559 -27.26409674 -33.66133197 -23.67826627
## [33,] -15.69948431  18.04289103 -24.71245564 -26.08749004 -28.60136362
## [34,] -15.97203190  17.50479258 -25.45669127 -29.15442411 -25.84129686
## [35,] -16.55435272  18.03986460 -26.34907934 -28.45923757 -29.19987437
## [36,] -15.88880721  17.60744756 -25.31346805 -28.38393322 -26.57464227
## [37,] -16.26782870  17.82601346 -25.87057148 -28.29168164 -28.20193006
## [38,] -17.56857271  18.27804681 -27.98840201 -31.44216443 -29.29556496
## [39,]  -7.18192008   8.31211896 -12.02214028 -13.63020980 -12.83186284
## [40,] -37.60059990  40.42918514 -58.32321303 -63.20467739 -63.39866002
## [41,]  -6.62966813   4.66001115 -11.33653988 -13.46884355 -11.31227311
## [42,] -19.21610398   9.29496346 -30.15001043 -32.37629786 -31.72766254
## [43,] -24.77859061  10.08945529 -37.36511932 -42.65829286 -39.67366730
## [44,] -22.44056648   9.58674454 -33.96638129 -36.65466521 -38.41544807
## [45,] -23.50764183   9.74009336 -35.76400576 -40.56522616 -37.43945119
## [46,] -23.23089008   9.78835050 -35.27635529 -39.00638187 -38.46324330
## [47,] -23.28716783   9.61624540 -35.42701200 -40.05209642 -37.26118381
## [48,] -23.16207719   9.66212414 -35.18689154 -39.51435252 -37.46170324
## [49,] -22.70151157   9.53118820 -34.41072606 -38.10464183 -37.45587687
## [50,] -23.96099839   9.89693734 -36.49881388 -41.19973303 -38.50048932
## [51,]  -9.55632821   3.36789876 -14.42842788 -15.92891176 -15.48293188
## [52,] -52.27325321  24.01020822 -79.25613704 -88.30706867 -86.30179515
## [53,]   5.48075774  -2.21487535   8.91269467  11.60257818   7.31881404
## [54,]   7.82505627  -7.04950919  13.22904979  15.13114490  13.51717029
## [55,]  10.79796230 -10.39806147  18.11665431  18.90775982  19.65631609
## [56,]   9.68261285  -9.33549131  16.57029992  19.20429186  15.39458018
## [57,]   9.67606187  -9.60313145  16.43940460  17.50640916  17.58541663
## [58,]   9.29612984  -9.36227386  15.79635096  17.50137664  15.93697859
## [59,]  10.12011358  -9.65513665  17.25448716  18.64726000  17.97954933
## [60,]   9.18145516  -9.30080060  15.69084274  17.18035313  16.02797748
## [61,]  10.72680782  -9.90612369  18.13724569  19.85843246  18.68604319
## [62,]  10.13315131  -9.65012532  17.22833125  18.99943350  17.48511032
## [63,]   3.18970141  -3.60033825   5.77654151   6.44613581   5.86605061
## [64,]  23.66963397 -23.35732514  39.60158092  42.15618815  41.88688202
## [65,]  -3.98133925   0.44505280  -5.70642567  -6.91622457  -5.16120685
## [66,]  -0.00930836   0.01249415  -0.01403533  -0.01474488  -0.01684455
##              [,21]        [,22]        [,23]        [,24]        [,25]
##  [1,]  17.80331477  17.51414120  18.02361343  17.61123333  17.58083482
##  [2,] -17.04464297 -17.12125305 -17.13198752 -17.04504583 -17.06412622
##  [3,] -18.06391259 -17.42503081 -18.36993574 -17.69105478 -17.62546414
##  [4,] -16.68962629 -17.23389431 -16.83246531 -16.89183688 -16.92570197
##  [5,] -18.41958706 -17.70430506 -18.81911933 -18.11768202 -17.96155655
##  [6,] -17.55021435 -17.69557366 -17.60584731 -17.28663437 -17.74846540
##  [7,] -17.97404899 -17.27300546 -18.31928430 -18.61991637 -16.44011338
##  [8,] -17.80769332 -17.60605072 -18.05417993 -16.33725125 -19.16443924
##  [9,] -17.89671207 -17.57133033 -18.07844094 -18.46270821 -16.64388335
## [10,] -17.53293533 -17.19054894 -17.90147620 -16.92848245 -18.18149725
## [11,] -10.60544188 -10.60163853 -10.61977644 -10.74923219 -10.17696948
## [12,] -35.35844102 -34.44462713 -35.93557944 -34.74155688 -34.86075493
## [13,]  -7.14103670  -7.06092470  -7.27856002  -6.85506963  -7.64754385
## [14,] -65.30902736 -64.78701270 -65.23691699 -64.69574415 -65.00414181
## [15,]  27.80541557  27.91085727  27.40650782  27.64512186  27.93797048
## [16,]  38.55257779  38.40367666  38.50589157  38.28509491  38.46668473
## [17,] -17.51125849 -17.38883102 -17.36631495 -17.32445912 -17.46148721
## [18,]  60.00313815  59.85975974  60.01712302  59.64072372  59.90455304
## [19,]  68.02507052  66.18488315  67.73095492  66.60934762  67.01823588
## [20,]  62.37332379  64.47515598  62.70139277  63.32422798  63.48447417
## [21,]  66.76365026  64.31789877  66.36327698  65.02940000  65.46681065
## [22,]  64.31789877  66.95415513  63.53556681  65.10576558  64.60171617
## [23,]  66.36327698  63.53556681  67.63114463  64.52620922  65.70231887
## [24,]  65.02940000  65.10576558  64.52620922  68.24194378  60.36474820
## [25,]  65.46681065  64.60171617  65.70231887  60.36474820  71.79176265
## [26,]  64.89840500  64.85431318  64.40209286  67.65983986  59.02170826
## [27,]  36.29380061  35.68065626  36.35803723  33.87075318  39.06672101
## [28,] 136.92949107 135.90095304 136.90675186 136.17398487 136.05002512
## [29,]  17.46825193  17.98416298  17.82572495  19.87801123  15.04635062
## [30,] -26.40973541 -25.16731374 -26.33168879 -25.47540360 -25.70333893
## [31,] -29.16148868 -32.14738611 -28.78237086 -30.69262052 -30.88226944
## [32,] -32.43626054 -27.35335855 -31.95694373 -29.49671059 -29.91055076
## [33,] -25.33052218 -27.96924434 -24.51163628 -26.33658579 -26.62661897
## [34,] -28.47831597 -25.65124649 -28.88436448 -27.37050535 -27.58140234
## [35,] -27.60721119 -29.77434288 -26.88025357 -26.95402658 -29.79843948
## [36,] -27.69471540 -26.83012391 -27.57104366 -30.88850275 -22.86248540
## [37,] -27.46256651 -28.16983427 -26.96560455 -23.16013834 -33.06743900
## [38,] -30.67420353 -30.58677165 -29.48003092 -31.25710035 -27.77096344
## [39,] -13.16986263 -12.88569926 -13.44091217 -12.87804592 -13.72618413
## [40,] -61.68377536 -63.06401520 -60.35782931 -61.80987617 -62.41693100
## [41,] -12.99594023 -12.65917596 -12.31740712 -12.40861723 -11.86724851
## [42,] -31.85282637 -31.99719321 -31.75825872 -31.79354014 -31.95172981
## [43,] -41.52714946 -40.87709389 -41.44129983 -40.95020817 -41.17750887
## [44,] -36.00672270 -37.52987274 -36.09344768 -36.78397545 -36.81825244
## [45,] -39.77082566 -38.58281673 -39.70219426 -38.87958239 -39.21537210
## [46,] -38.00334353 -40.13629810 -37.10698666 -38.78100108 -37.93345276
## [47,] -39.25861650 -37.37919636 -40.26877092 -37.77848513 -39.61217103
## [48,] -38.69048171 -38.59341733 -38.44356721 -43.69283565 -31.47075289
## [49,] -37.14386362 -37.27181556 -37.38615777 -27.98832763 -50.63673109
## [50,] -40.37055637 -40.49126744 -39.38170305 -45.20340700 -30.43270779
## [51,] -15.61357752 -15.29933490 -15.93520451 -11.70271202 -22.01499834
## [52,] -86.11827019 -86.31861444 -87.13466632 -87.36493825 -86.40825190
## [53,]  11.44912228  10.09904948   9.64084245   5.39726374  15.02664978
## [54,]  14.64433369  14.16627605  14.58889956  14.26811511  14.37755040
## [55,]  18.64318544  19.36726906  18.51098968  18.93456854  19.03948145
## [56,]  18.59821537  16.71531820  18.40289474  17.51609880  17.71306545
## [57,]  17.03092336  17.79182045  16.76295745  17.23212924  17.31902664
## [58,]  17.11107772  15.52970328  17.48587162  16.43748560  16.93895550
## [59,]  18.11436480  18.92979575  17.76060033  17.77148216  18.62402530
## [60,]  16.77022501  16.25276897  16.87585446  19.64085210  12.68457350
## [61,]  19.32621772  19.33002052  18.75376887  14.30616375  24.99970665
## [62,]  18.55253343  18.50987137  17.90545181  20.67036742  13.86224634
## [63,]   6.23915780   5.92038559   6.36998799   4.39538166   8.97469823
## [64,]  41.05773107  41.48777587  41.21344736  41.74528764  41.51261134
## [65,]  -6.92653000  -6.39025396  -6.04203296  -4.24864848  -8.42479385
## [66,]  -0.01430868  -0.01546855  -0.01461416  -0.01567292  -0.01509746
##              [,26]         [,27]         [,28]         [,29]         [,30]
##  [1,]  17.66178815  10.553316210   34.93153964    5.63622874   0.060843800
##  [2,] -17.03299336  -9.977748804  -34.38632113   -5.49563391  -0.848096827
##  [3,] -17.78361907 -10.745422249  -34.75744987   -5.57122380   0.368597037
##  [4,] -16.90956255 -10.027585780  -33.70037443   -5.58185142  -0.992421705
##  [5,] -18.17838954 -10.865085224  -35.74475090   -5.73948627   0.441222068
##  [6,] -17.35785509 -10.561538951  -34.74377255   -5.49253627  -0.248511075
##  [7,] -18.53912082 -10.044115868  -35.16839947   -6.40123062   0.086034082
##  [8,] -16.62418254 -11.320383406  -34.72374011   -4.53689579  -0.086788667
##  [9,] -18.49176181 -10.258619958  -35.03654249   -5.88253082  -0.011087859
## [10,] -15.20933589 -10.973416695  -35.21641399   -6.46356735  -0.170253016
## [11,] -11.77831563  -5.083014189  -21.15762654   -4.13704245  -0.070408125
## [12,] -34.44365682 -21.885303756  -68.42048648  -10.58685978  -0.002670521
## [13,]  -5.35628544  -6.253987318  -14.44424832   -0.67266566   0.070414230
## [14,] -64.49561408 -35.902298193 -136.10893269  -17.84116589  25.620318019
## [15,]  27.36581648  14.348524341   60.90673860    6.78185902 -24.547010583
## [16,]  38.15438678  21.186073239   80.64405249   10.58849868 -14.390990568
## [17,] -17.20615019  -9.345906204  -37.24881552   -4.52024050  12.687787166
## [18,]  59.44637229  33.407297327  124.62326971   17.00552532 -23.018509785
## [19,]  66.45346400  36.984090160  140.53728633   17.77797200 -27.420608063
## [20,]  63.02479044  34.873144321  133.07017415   18.27351545 -23.602462306
## [21,]  64.89840500  36.293800607  136.92949107   17.46825193 -26.409735414
## [22,]  64.85431318  35.680656259  135.90095304   17.98416298 -25.167313743
## [23,]  64.40209286  36.358037230  136.90675186   17.82572495 -26.331688786
## [24,]  67.65983986  33.870753183  136.17398487   19.87801123 -25.475403600
## [25,]  59.02170826  39.066721007  136.05002512   15.04635062 -25.703338935
## [26,]  74.69877594  29.621938546  135.98116099   23.17144040 -25.412992474
## [27,]  29.62193855  30.672341605   69.99708674   -5.66082913 -13.934494810
## [28,] 135.98116099  69.997086740  294.12184323   52.71604992 -54.679719445
## [29,]  23.17144040  -5.660829129   52.71604992   43.43433194  -5.874538583
## [30,] -25.41299247 -13.934494810  -54.67971945   -5.87453858  27.435195518
## [31,] -30.19299554 -15.321041970  -68.12028897   -8.63029219  22.489583841
## [32,] -29.39982551 -16.165008927  -64.17043087   -6.12025683  29.451550502
## [33,] -25.97446431 -13.212920011  -58.56535642   -7.07972581  22.381479212
## [34,] -27.07349910 -14.376752893  -60.29856611   -6.40967453  25.053119582
## [35,] -26.93542530 -15.132827288  -61.56958074   -6.19940357  24.433899938
## [36,] -29.34052557 -11.378719661  -61.98383587  -12.13908573  24.622820629
## [37,] -25.82813159 -15.911927107  -59.92477940   -4.09885627  24.280531010
## [38,] -37.12454117 -16.896143151  -57.50315840    7.65745138  26.023054323
## [39,] -10.01547941  -9.670398091  -31.51141197   -6.42735005  14.059639407
## [40,] -62.22084388 -31.707870241 -133.40178807   -8.89271198  48.649593943
## [41,] -16.33018863  -5.151855532  -17.40125707   10.41553423  11.221204914
## [42,] -31.64240570 -17.596796291  -66.91010467   -9.07965951  10.417643337
## [43,] -40.83591704 -22.702607048  -86.32086970  -11.13215398  16.165790163
## [44,] -36.55739667 -20.042528209  -77.59288220  -10.75026167  12.303873375
## [45,] -38.80025336 -21.713063768  -81.92942882  -10.48639325  15.301696166
## [46,] -38.66477091 -21.064277623  -80.36281336  -10.26648727  14.061285773
## [47,] -37.64513880 -21.576892720  -81.50686962  -11.05118884  14.972346101
## [48,] -43.12918848 -18.652320288  -80.71433747  -12.51736003  14.498618930
## [49,] -27.03033349 -23.538727097  -81.79748900  -13.60832860  13.637437702
## [50,] -58.51989630 -16.147884623  -77.38533429   -4.55176510  15.397177041
## [51,]  -2.61317826 -21.826311040  -30.26276037    8.80626515   4.935636611
## [52,] -84.55102387 -32.580141047 -211.34257776  -82.85952949  33.919579251
## [53,]   3.25538851  40.116813606  -29.74067996 -107.80247223  -5.719513426
## [54,]  14.21763761   7.838274940   30.42825188    3.56858411 -12.181178106
## [55,]  18.76093219  10.061189498   40.79808615    5.21655874 -12.564400882
## [56,]  17.42415163   9.633534390   37.63654240    4.33759582 -13.862482795
## [57,]  17.10979946   9.202552702   37.01886729    4.58486160 -12.289285567
## [58,]  16.25369501   8.926749649   36.07102720    4.64151573 -12.286602764
## [59,]  17.88170623  10.203493299   38.26136752    3.62628210 -13.233171515
## [60,]  18.13801297   5.715229155   38.64626988   11.90769142 -12.374036210
## [61,]  17.66804616  16.047903001   33.95644894  -10.85481366 -13.522390129
## [62,]  27.30207945   6.885576344   35.64476051    1.23023729 -13.280353336
## [63,]   0.05255964  10.025298941   11.60370991   -5.66569796  -5.837799736
## [64,]  40.15133887  14.539993120  103.25504640   40.06753197 -27.551360776
## [65,]  -3.33178523 -20.975360846   11.93592559   53.01062448   0.532101724
## [66,]  -0.01408504   0.002209006   -0.05057119   -0.03768717   0.015376186
##              [,31]        [,32]        [,33]        [,34]        [,35]
##  [1,]   8.03471684   2.16466871   7.20425838   4.93159528   5.56376923
##  [2,]  -1.01602835  -0.64628252  -1.76839285  -1.27958395  -1.16338863
##  [3,] -12.47647017  -2.29441716 -10.40797502  -6.70572910  -8.02276622
##  [4,]  -4.59097889  -4.35143974  -4.42689673  -4.75780405  -3.67755348
##  [5,]  -9.80347932  -1.27075065  -9.11763426  -4.73224151  -6.80523620
##  [6,]  -6.87015513  -2.94203939  -5.75431448  -5.72173011  -4.19645570
##  [7,]  -8.90227740  -1.49882601  -8.24251818  -4.29192277  -7.35085002
##  [8,]  -7.60753377  -2.40867410  -6.69371365  -5.18528365  -4.33838798
##  [9,]  -8.25608218  -1.99745179  -7.45709930  -4.79160279  -6.18907544
## [10,]  -7.76274320  -2.27790963  -6.97693326  -4.91201574  -5.24951931
## [11,]  -2.34157591  -0.53848207  -2.39674699  -1.52170876  -1.84672781
## [12,] -22.13874848  -6.08000554 -19.13310596 -13.25503392 -14.99628457
## [13,]   0.80848339   0.51305845   0.36521436   0.34247510   0.79273646
## [14,]  30.65528907  29.83931898  26.35911590  27.54819947  28.18118426
## [15,] -44.67208875 -34.24964186 -39.57868166 -36.86082690 -38.52195074
## [16,] -18.70670720 -16.76288293 -15.69948431 -15.97203190 -16.55435272
## [17,]  20.03422469  17.22241559  18.04289103  17.50479258  18.03986460
## [18,] -28.94351256 -27.26409674 -24.71245564 -25.45669127 -26.34907933
## [19,] -29.04061721 -33.66133197 -26.08749004 -29.15442411 -28.45923757
## [20,] -34.32634244 -23.67826627 -28.60136362 -25.84129686 -29.19987436
## [21,] -29.16148868 -32.43626054 -25.33052217 -28.47831596 -27.60721119
## [22,] -32.14738611 -27.35335854 -27.96924434 -25.65124649 -29.77434288
## [23,] -28.78237086 -31.95694373 -24.51163628 -28.88436448 -26.88025357
## [24,] -30.69262052 -29.49671059 -26.33658579 -27.37050535 -26.95402658
## [25,] -30.88226944 -29.91055076 -26.62661897 -27.58140234 -29.79843948
## [26,] -30.19299554 -29.39982551 -25.97446431 -27.07349909 -26.93542530
## [27,] -15.32104197 -16.16500893 -13.21292001 -14.37675289 -15.13282729
## [28,] -68.12028897 -64.17043087 -58.56535642 -60.29856611 -61.56958074
## [29,]  -8.63029219  -6.12025683  -7.07972581  -6.40967453  -6.19940357
## [30,]  22.48958384  29.45155050  22.38147921  25.05311958  24.43389994
## [31,]  62.16679848  29.63235528  50.79935123  41.12393841  46.68233858
## [32,]  29.63235528  48.83387104  28.93357331  38.53223339  31.42583622
## [33,]  50.79935123  28.93357331  45.83935895  34.76963648  42.53046388
## [34,]  41.12393841  38.53223339  34.76963648  41.27890092  33.66635924
## [35,]  46.68233858  31.42583622  42.53046388  33.66635924  43.33916327
## [36,]  43.12570994  35.67221709  37.67123195  38.25990781  34.49488601
## [37,]  45.48966649  32.95835473  40.61061108  35.71596152  41.35451094
## [38,]  43.60409880  36.27344769  38.90170109  37.59001656  38.52225340
## [39,]  18.16697716  16.55581124  16.89893584  16.35976697  16.94636965
## [40,] 109.48841503  74.97315811  94.80850603  85.46481819  90.24929142
## [41,]   4.42289499  10.71270258   5.10231437   7.76496518   6.26668502
## [42,]  19.67471271  14.88970663  16.37958891  15.58350880  16.45253972
## [43,]  18.29719954  18.72480770  15.51524317  16.74092535  16.78624738
## [44,]  22.79035489  12.09863889  18.31723844  15.52726888  17.71004477
## [45,]  16.91976090  18.64814363  14.36203654  16.28575137  15.93368510
## [46,]  20.15563531  14.93033039  17.42623295  14.39282303  18.19147048
## [47,]  16.93486379  18.41161864  13.83566161  17.09661115  15.32825917
## [48,]  18.19458225  17.23410192  15.12728123  16.33140807  14.24323757
## [49,]  19.73429140  15.12973965  16.66877447  15.02454076  19.96696112
## [50,]  17.43731772  17.94918439  14.91064508  15.98395765  15.32223947
## [51,]   6.56926854   5.60204562   5.27069570   5.46181157   6.43572779
## [52,]  50.93205378  39.79060868  42.73331689  40.62205256  42.47366761
## [53,]   2.54638401  -7.48225739   1.31528250  -3.21690867  -2.98953411
## [54,] -12.22858729 -14.14571998 -11.72840442 -12.62184888 -12.47945944
## [55,] -25.05668920 -16.25886252 -21.41276458 -18.91347443 -20.69383508
## [56,] -15.96032150 -21.91579111 -15.13720636 -18.49845603 -15.80529142
## [57,] -21.48378194 -15.58328679 -19.64195064 -16.59272641 -19.27040297
## [58,] -19.56761871 -18.54306999 -16.74795839 -19.62347972 -16.16014656
## [59,] -19.89748634 -16.59273901 -18.56036136 -16.04363672 -19.73100880
## [60,] -19.43898511 -17.49491845 -17.19062619 -18.03120679 -15.58561759
## [61,] -20.18656443 -17.75784015 -18.51731509 -17.40231347 -20.76409680
## [62,] -19.31441307 -17.97451075 -17.58487539 -17.58729207 -17.46903193
## [63,]  -6.48118715  -6.90321581  -6.17346256  -6.43917137  -6.81037022
## [64,] -54.23857210 -40.02617965 -47.63719173 -43.74039611 -45.73811714
## [65,]  -1.03246120   1.92395552  -1.05197250   0.55907492   0.89062663
## [66,]   0.03826785   0.02197809   0.03296227   0.02746149   0.02996011
##              [,36]        [,37]         [,38]        [,39]         [,40]
##  [1,]   5.35973243   5.54308261   4.363227292   1.86959790   14.39762841
##  [2,]  -1.39305210  -1.24279819  -0.582645448  -0.97138757   -2.08933815
##  [3,]  -7.40930013  -7.89532758  -6.320637158  -2.37490859  -20.98970934
##  [4,]  -4.52772769  -4.01939026  -3.447528751  -1.71010980  -10.29184268
##  [5,]  -5.70727984  -6.48015276  -4.889216144  -1.96746077  -16.67132173
##  [6,]  -5.51943203  -4.58061501  -4.001863519  -1.75963240  -12.97243207
##  [7,]  -3.93133204  -7.30372204  -4.589196046  -1.87597700  -15.16503043
##  [8,]  -7.45481633  -3.42085953  -4.183148594  -1.89143480  -14.19993422
##  [9,]  -4.20563411  -7.98846944  -3.463293774  -2.49755520  -13.59914256
## [10,]  -5.33771465  -4.88327389  -9.062320116   0.74904233  -16.30587248
## [11,]  -1.54543770  -2.20370337   1.010145041  -3.40531628    0.41176209
## [12,] -14.52884434 -14.70991649 -12.169680475  -2.81449489  -47.84267888
## [13,]  -0.06910155   1.24816160  -0.702054674   4.04274914  -13.30720968
## [14,]  27.27614210  27.74011901  30.175050178  13.10173031   62.15415284
## [15,] -37.36592529 -38.04647131 -38.244358116 -16.69820624  -88.75731067
## [16,] -15.88880721 -16.26782870 -17.568572714  -7.18192008  -37.60059990
## [17,]  17.60744756  17.82601346  18.278046814   8.31211896   40.42918514
## [18,] -25.31346805 -25.87057148 -27.988402008 -12.02214028  -58.32321303
## [19,] -28.38393322 -28.29168164 -31.442164429 -13.63020979  -63.20467739
## [20,] -26.57464227 -28.20193006 -29.295564962 -12.83186284  -63.39866002
## [21,] -27.69471540 -27.46256651 -30.674203528 -13.16986263  -61.68377535
## [22,] -26.83012391 -28.16983427 -30.586771652 -12.88569926  -63.06401520
## [23,] -27.57104366 -26.96560455 -29.480030921 -13.44091217  -60.35782931
## [24,] -30.88850275 -23.16013834 -31.257100352 -12.87804592  -61.80987617
## [25,] -22.86248540 -33.06743900 -27.770963444 -13.72618413  -62.41693100
## [26,] -29.34052557 -25.82813159 -37.124541173 -10.01547941  -62.22084388
## [27,] -11.37871966 -15.91192711 -16.896143150  -9.67039809  -31.70787024
## [28,] -61.98383587 -59.92477940 -57.503158398 -31.51141197 -133.40178806
## [29,] -12.13908573  -4.09885627   7.657451380  -6.42735005   -8.89271198
## [30,]  24.62282063  24.28053101  26.023054323  14.05963941   48.64959394
## [31,]  43.12570994  45.48966649  43.604098800  18.16697716  109.48841503
## [32,]  35.67221709  32.95835473  36.273447686  16.55581124   74.97315811
## [33,]  37.67123195  40.61061108  38.901701095  16.89893584   94.80850603
## [34,]  38.25990781  35.71596152  37.590016560  16.35976697   85.46481819
## [35,]  34.49488601  41.35451094  38.522253398  16.94636965   90.24929142
## [36,]  44.70282163  30.21654985  34.939680362  17.93303979   86.80509787
## [37,]  30.21654985  53.46463852  37.079425170  17.94145253   88.95305539
## [38,]  34.93968036  37.07942517  72.739550438   0.10693023   89.57917630
## [39,]  17.93303979  17.94145253   0.106930230  22.34840335   28.40118315
## [40,]  86.80509787  88.95305539  89.579176299  28.40118315  258.07233585
## [41,]   5.67885870   5.35642175  21.325373411 -14.71128528   81.44604186
## [42,]  15.67939365  16.16909819  16.942290845   6.26968488   39.50252998
## [43,]  16.42743341  16.60225997  18.256840530   7.66471743   37.71098191
## [44,]  16.08483019  17.14467772  17.425024491   7.00859620   40.64535131
## [45,]  15.81125248  15.85258266  17.603594889   7.25249427   36.19050859
## [46,]  15.91024376  16.52880930  18.742687546   6.85827796   38.93756797
## [47,]  15.41981797  16.25585812  15.959813116   7.67783639   35.72696826
## [48,]  22.07494519   8.26407460  20.368091108   6.62877068   36.45158137
## [49,]   6.45709926  31.15142830   3.901874407  11.49674099   38.46242694
## [50,]  16.73209011  14.84637845  51.168839942  -6.00300966   33.98497455
## [51,]   2.78427593   7.01738882  -8.509582665  10.83191002   20.20305797
## [52,]  49.39016484  39.77581848   3.397067174  42.69021113   45.37736097
## [53,]  13.88813691  -6.87803140 -62.877547093  33.82436811  -97.08399657
## [54,] -12.46586124 -12.38703992 -13.155175656  -6.85413089  -25.57130193
## [55,] -19.48767922 -20.21361140 -20.159009323  -8.89161175  -46.77357983
## [56,] -17.36764896 -16.42430945 -17.660630513  -8.30059494  -36.95677240
## [57,] -17.54205147 -18.47205203 -18.529204150  -8.34590616  -41.74671070
## [58,] -18.06090610 -17.33714850 -17.488523021  -8.34901197  -40.05200990
## [59,] -16.36986628 -18.77539835 -19.347345857  -8.05572155  -40.25636496
## [60,] -23.17923483 -11.82691876 -12.030150664 -10.93736459  -38.38859504
## [61,]  -9.34164576 -29.76445980 -32.605243914  -2.65201397  -43.71316564
## [62,] -17.67492587 -16.88511454 -38.084016874   0.18321138  -37.78649364
## [63,]  -5.24222638  -7.36366916   1.398571476  -9.14093512  -15.96200479
## [64,] -48.84247717 -44.12224088 -23.363052410 -31.04289719  -90.13795065
## [65,]  -8.05601492   3.01203601  33.190702705 -16.86347636   33.71819053
## [66,]   0.03365210   0.02844412   0.003357603   0.02499368    0.05096532
##               [,41]        [,42]         [,43]        [,44]         [,45]
##  [1,]   -2.87192569  -7.61577417 -10.814243986  -8.75297809 -10.431792541
##  [2,]    1.49957948   8.62797251  10.349967628   9.50826202  10.003006379
##  [3,]    3.63382737   6.99019434  10.878533532   7.98912305  10.573715188
##  [4,]    1.99371167   7.82187504  10.229020981   9.48226989   9.738729826
##  [5,]    3.41278910   7.53596950  11.154180570   8.49766975  10.862191832
##  [6,]    2.48350564   7.73011390  10.703403141   8.99360135  10.272885212
##  [7,]    3.44277844   7.50904774  10.865701326   8.51464154  10.529351827
##  [8,]    2.24266129   7.69702376  10.837176905   8.88481913  10.444165528
##  [9,]    5.15444444   7.61119543  10.857822762   8.72209916  10.478794166
## [10,]   -2.86598201   7.54945222  10.655336073   8.70254305  10.271923858
## [11,]   10.06810580   5.01460982   6.541099160   5.65348768   6.306293142
## [12,]   -6.33359009  13.92141263  21.238443316  16.25264741  20.499552419
## [13,]  -28.98664946   3.77295505   4.417969276   4.11668228   4.273640481
## [14,]   12.55307799  31.85833668  41.074560325  36.74848663  39.070637684
## [15,]   -6.99960316 -16.07673212 -16.662289102 -16.81771664 -15.954868259
## [16,]   -6.62966813 -19.21610398 -24.778590611 -22.44056648 -23.507641832
## [17,]    4.66001115   9.29496346  10.089455289   9.58674454   9.740093363
## [18,]  -11.33653988 -30.15001043 -37.365119323 -33.96638129 -35.764005760
## [19,]  -13.46884355 -32.37629786 -42.658292863 -36.65466521 -40.565226162
## [20,]  -11.31227311 -31.72766254 -39.673667301 -38.41544807 -37.439451187
## [21,]  -12.99594023 -31.85282637 -41.527149461 -36.00672270 -39.770825664
## [22,]  -12.65917596 -31.99719321 -40.877093887 -37.52987274 -38.582816735
## [23,]  -12.31740712 -31.75825872 -41.441299835 -36.09344768 -39.702194260
## [24,]  -12.40861723 -31.79354014 -40.950208171 -36.78397545 -38.879582390
## [25,]  -11.86724851 -31.95172981 -41.177508868 -36.81825244 -39.215372097
## [26,]  -16.33018863 -31.64240570 -40.835917042 -36.55739667 -38.800253359
## [27,]   -5.15185553 -17.59679629 -22.702607048 -20.04252821 -21.713063768
## [28,]  -17.40125707 -66.91010467 -86.320869702 -77.59288220 -81.929428824
## [29,]   10.41553423  -9.07965951 -11.132153982 -10.75026167 -10.486393255
## [30,]   11.22120491  10.41764334  16.165790163  12.30387337  15.301696166
## [31,]    4.42289499  19.67471271  18.297199540  22.79035489  16.919760905
## [32,]   10.71270258  14.88970663  18.724807696  12.09863889  18.648143626
## [33,]    5.10231437  16.37958891  15.515243170  18.31723844  14.362036541
## [34,]    7.76496518  15.58350880  16.740925350  15.52726888  16.285751365
## [35,]    6.26668502  16.45253972  16.786247385  17.71004477  15.933685098
## [36,]    5.67885870  15.67939365  16.427433405  16.08483019  15.811252480
## [37,]    5.35642175  16.16909819  16.602259967  17.14467772  15.852582655
## [38,]   21.32537341  16.94229085  18.256840530  17.42502449  17.603594890
## [39,]  -14.71128528   6.26968488   7.664717431   7.00859620   7.252494272
## [40,]   81.44604187  39.50252998  37.710981907  40.64535131  36.190508593
## [41,]  143.87797531   4.34537149   7.595755554   5.20267164   7.243447596
## [42,]    4.34537149  17.51161339  20.022597599  19.16834549  19.229383930
## [43,]    7.59575555  20.02259760  26.769806125  23.43672031  25.340354110
## [44,]    5.20267164  19.16834549  23.436720306  23.66425720  21.932717218
## [45,]    7.24344760  19.22938393  25.340354110  21.93271722  24.253887277
## [46,]    6.80100191  19.38251114  24.642589155  23.05271215  23.146342617
## [47,]    6.46055748  19.08775832  25.069783644  21.89052058  24.020003032
## [48,]    6.05239423  19.14595464  24.788919663  22.29303589  23.502874848
## [49,]    3.62622797  18.95656541  24.130663554  22.42090769  22.849267647
## [50,]   12.03271289  19.63371049  25.733597610  22.64422547  24.483688379
## [51,]    8.75236154   7.91598023  10.192425907   9.31673877   9.693314787
## [52,]  -92.98293173  43.96938755  55.521036817  51.93998928  52.365595044
## [53,] -191.41332815  -3.69222738  -6.450258832  -3.02987443  -6.620476406
## [54,]   -5.29832316  -6.38308498  -8.539266817  -7.04530839  -8.163185231
## [55,]   -4.18682543 -10.64066458 -10.984819144 -11.61809695 -10.446866435
## [56,]   -5.47873306  -8.88485303 -10.523977032  -8.07264290 -10.312984315
## [57,]   -4.33828599  -9.42411585  -9.909898177 -10.06612478  -9.457736709
## [58,]   -4.25579278  -9.02155689  -9.730458662  -9.11760535  -9.432003909
## [59,]   -5.38016426  -9.52597304 -10.532020476  -9.88398954 -10.184947529
## [60,]   -0.61256812  -8.86435149  -9.568612788  -9.03618100  -9.244604386
## [61,]  -13.54872197 -10.05797987 -11.226757987 -10.36914758 -10.890231553
## [62,]   -7.35639162  -9.53051373 -10.632591828  -9.68357091 -10.303942406
## [63,]   -4.22522925  -2.77182560  -3.437563280  -2.93011223  -3.318016365
## [64,]   29.97960376 -23.91665029 -24.088088747 -24.76985729 -23.149958374
## [65,]   66.43530900   3.25250821   4.380629757   3.04513415   4.425795757
## [66,]   -0.04157824   0.01033157   0.008997963   0.01144503   0.008324022
##               [,46]         [,47]         [,48]        [,49]        [,50]
##  [1,]  -9.703240201 -10.344041265 -10.018976568  -9.38724293 -10.73455470
##  [2,]   9.851933770   9.943968854   9.856687034   9.59190963  10.30103138
##  [3,]   9.383388977  10.470935370   9.965600415   9.03982573  10.85881561
##  [4,]   9.840229917   9.630485203   9.582264479   9.47749710  10.17048367
##  [5,]   9.593810043  10.878299199  10.372929858   9.28934986  11.12118330
##  [6,]   9.974975619  10.046678872   9.585866449  10.04228922  10.38332549
##  [7,]   9.462673694  10.451442528  11.541778526   6.91720245  11.88421230
##  [8,]   9.745567719  10.506599633   8.053162176  12.53849870   9.44786014
##  [9,]   9.771030368  10.257279106  11.303350306   6.96696012  11.90057852
## [10,]   9.438593807  10.408699494   9.199919506  11.45790957   5.27823975
## [11,]   6.097732384   6.184556585   6.352243608   5.23450189   8.44053926
## [12,]  18.597777274  20.310123809  19.505001433  17.76177184  21.57783948
## [13,]   4.196542917   4.303925136   4.120583140   4.27301186   3.83546072
## [14,]  38.346384963  38.671975751  38.370850972  37.42437386  39.82322697
## [15,] -16.588608443 -15.763397521 -16.073447960 -16.23353606 -16.15069694
## [16,] -23.230890076 -23.287167833 -23.162077186 -22.70151157 -23.96099839
## [17,]   9.788350501   9.616245400   9.662124138   9.53118820   9.89693734
## [18,] -35.276355289 -35.427012002 -35.186891536 -34.41072606 -36.49881388
## [19,] -39.006381872 -40.052096425 -39.514352517 -38.10464183 -41.19973303
## [20,] -38.463243305 -37.261183807 -37.461703239 -37.45587687 -38.50048932
## [21,] -38.003343531 -39.258616502 -38.690481714 -37.14386362 -40.37055637
## [22,] -40.136298105 -37.379196359 -38.593417327 -37.27181556 -40.49126745
## [23,] -37.106986661 -40.268770922 -38.443567213 -37.38615777 -39.38170305
## [24,] -38.781001078 -37.778485134 -43.692835647 -27.98832763 -45.20340700
## [25,] -37.933452764 -39.612171026 -31.470752885 -50.63673109 -30.43270779
## [26,] -38.664770914 -37.645138799 -43.129188484 -27.03033349 -58.51989630
## [27,] -21.064277623 -21.576892720 -18.652320288 -23.53872710 -16.14788462
## [28,] -80.362813358 -81.506869620 -80.714337469 -81.79748900 -77.38533429
## [29,] -10.266487275 -11.051188838 -12.517360030 -13.60832860  -4.55176510
## [30,]  14.061285774  14.972346102  14.498618930  13.63743770  15.39717704
## [31,]  20.155635308  16.934863789  18.194582250  19.73429140  17.43731772
## [32,]  14.930330386  18.411618643  17.234101916  15.12973965  17.94918439
## [33,]  17.426232953  13.835661612  15.127281228  16.66877447  14.91064508
## [34,]  14.392823031  17.096611147  16.331408066  15.02454076  15.98395765
## [35,]  18.191470478  15.328259168  14.243237571  19.96696112  15.32223947
## [36,]  15.910243758  15.419817965  22.074945193   6.45709926  16.73209011
## [37,]  16.528809301  16.255858121   8.264074602  31.15142830  14.84637845
## [38,]  18.742687546  15.959813116  20.368091108   3.90187441  51.16883994
## [39,]   6.858277965   7.677836387   6.628770678  11.49674099  -6.00300966
## [40,]  38.937567969  35.726968261  36.451581373  38.46242694  33.98497455
## [41,]   6.801001913   6.460557482   6.052394226   3.62622797  12.03271289
## [42,]  19.382511135  19.087758324  19.145954641  18.95656541  19.63371049
## [43,]  24.642589155  25.069783644  24.788919663  24.13066355  25.73359761
## [44,]  23.052712146  21.890520583  22.293035887  22.42090769  22.64422547
## [45,]  23.146342617  24.020003032  23.502874848  22.84926765  24.48368838
## [46,]  25.052162193  21.903610604  23.730232074  21.73262646  25.34522080
## [47,]  21.903610604  24.837660870  22.334727765  24.54863206  22.14182509
## [48,]  23.730232074  22.334727765  32.072068256   6.36420077  33.62023222
## [49,]  21.732626460  24.548632055   6.364200769  55.91672810  -2.16823717
## [50,]  25.345220796  22.141825091  33.620232216  -2.16823717  78.49981950
## [51,]   8.944644424  10.537791848   3.098070939  23.10606947 -20.77893847
## [52,]  51.281708960  53.882527278  51.950341177  66.08810630  22.82822678
## [53,]  -6.899391624  -3.755132688  -1.380147747  10.38804987 -37.16814659
## [54,]  -7.720982674  -8.020732397  -7.859531075  -7.49765035  -8.25360237
## [55,] -11.226839502 -10.354851864 -10.668049915 -10.88431766 -10.74569227
## [56,]  -8.985150137 -10.255627357  -9.827477880  -9.09713615 -10.04342339
## [57,] -10.277107758  -9.159354255  -9.552225731  -9.69256548  -9.86551384
## [58,]  -8.179653576 -10.074799679  -9.151721757  -9.47041523  -8.63199491
## [59,] -10.976952955  -9.663693757  -9.493845205 -10.56827714 -11.09231534
## [60,]  -9.102176095  -9.045691863 -14.308677957  -2.05689860  -7.38999356
## [61,] -10.927029730 -10.766676593  -3.085776297 -20.38663421 -20.18943496
## [62,] -10.904544029  -9.079590935 -14.694870707   2.93394215 -38.68834853
## [63,]  -2.874991943  -3.681572265  -0.364810255  -9.22042783  11.24106327
## [64,] -23.477361089 -23.964065156 -23.088927344 -31.84292055  -5.58951829
## [65,]   4.852902183   2.895239551   2.680477931  -5.94825989  24.87242055
## [66,]   0.009221083   0.009430343   0.008735318   0.01901929  -0.01127653
##               [,51]        [,52]        [,53]         [,54]        [,55]
##  [1,] -4.468151e+00  -19.6581930    4.3728844   1.535423525   0.36242681
##  [2,]  4.384611e+00   20.7497819   -2.9842503  -1.452910157  -2.30463189
##  [3,]  4.416907e+00   18.5844838   -5.3155287  -1.598155930   0.83012982
##  [4,]  4.379749e+00   19.7524550   -3.2987444  -1.226075491  -1.19943882
##  [5,]  4.516261e+00   19.7204300   -4.9162047  -1.701958049   0.04752874
##  [6,]  4.640312e+00   19.7778284   -4.2859316  -1.484463653  -0.68381328
##  [7,]  3.583303e+00   20.0126484   -3.0345350  -1.574396291  -0.11802947
##  [8,]  5.566855e+00   18.9686166   -6.5295178  -1.537908023  -0.49442410
##  [9,]  4.001035e+00   18.4377868   -6.2023272  -1.552794035  -0.30836648
## [10,]  6.729558e+00   25.6308177    4.6693778  -1.489360708  -0.40212028
## [11,]  1.402540e+00    7.3993771  -10.0184465  -0.896648322  -0.86326455
## [12,]  8.181551e+00   44.2643391    3.0883687  -3.090650723   0.92888871
## [13,]  1.914217e+00   25.1613067   26.2609807  -0.686695934  -1.28462964
## [14,]  1.562266e+01   86.4907388  -10.0043445 -14.332977406 -18.95233797
## [15,] -5.467127e+00  -42.0093453    1.8975473  12.485634266  20.00480195
## [16,] -9.556328e+00  -52.2732532    5.4807577   7.825056271  10.79796230
## [17,]  3.367899e+00   24.0102082   -2.2148754  -7.049509189 -10.39806147
## [18,] -1.442843e+01  -79.2561370    8.9126947  13.229049791  18.11665431
## [19,] -1.592891e+01  -88.3070687   11.6025782  15.131144901  18.90775982
## [20,] -1.548293e+01  -86.3017951    7.3188140  13.517170292  19.65631609
## [21,] -1.561358e+01  -86.1182702   11.4491223  14.644333689  18.64318544
## [22,] -1.529933e+01  -86.3186144   10.0990495  14.166276049  19.36726906
## [23,] -1.593520e+01  -87.1346663    9.6408425  14.588899561  18.51098968
## [24,] -1.170271e+01  -87.3649383    5.3972637  14.268115109  18.93456854
## [25,] -2.201500e+01  -86.4082519   15.0266498  14.377550404  19.03948145
## [26,] -2.613178e+00  -84.5510239    3.2553885  14.217637609  18.76093219
## [27,] -2.182631e+01  -32.5801410   40.1168136   7.838274940  10.06118950
## [28,] -3.026276e+01 -211.3425778  -29.7406800  30.428251876  40.79808615
## [29,]  8.806265e+00  -82.8595295 -107.8024722   3.568584110   5.21655874
## [30,]  4.935637e+00   33.9195793   -5.7195134 -12.181178106 -12.56440088
## [31,]  6.569269e+00   50.9320538    2.5463840 -12.228587290 -25.05668920
## [32,]  5.602046e+00   39.7906087   -7.4822574 -14.145719980 -16.25886252
## [33,]  5.270696e+00   42.7333169    1.3152825 -11.728404422 -21.41276458
## [34,]  5.461812e+00   40.6220526   -3.2169087 -12.621848885 -18.91347443
## [35,]  6.435728e+00   42.4736676   -2.9895341 -12.479459442 -20.69383508
## [36,]  2.784276e+00   49.3901648   13.8881369 -12.465861243 -19.48767922
## [37,]  7.017389e+00   39.7758185   -6.8780314 -12.387039915 -20.21361140
## [38,] -8.509583e+00    3.3970672  -62.8775471 -13.155175655 -20.15900932
## [39,]  1.083191e+01   42.6902111   33.8243681  -6.854130888  -8.89161175
## [40,]  2.020306e+01   45.3773610  -97.0839966 -25.571301933 -46.77357983
## [41,]  8.752362e+00  -92.9829317 -191.4133281  -5.298323156  -4.18682543
## [42,]  7.915980e+00   43.9693875   -3.6922274  -6.383084976 -10.64066458
## [43,]  1.019243e+01   55.5210368   -6.4502588  -8.539266816 -10.98481914
## [44,]  9.316739e+00   51.9399893   -3.0298744  -7.045308391 -11.61809695
## [45,]  9.693315e+00   52.3655950   -6.6204764  -8.163185231 -10.44686643
## [46,]  8.944644e+00   51.2817090   -6.8993916  -7.720982674 -11.22683950
## [47,]  1.053779e+01   53.8825273   -3.7551327  -8.020732397 -10.35485186
## [48,]  3.098071e+00   51.9503412   -1.3801477  -7.859531075 -10.66804991
## [49,]  2.310607e+01   66.0881063   10.3880499  -7.497650354 -10.88431766
## [50,] -2.077894e+01   22.8282268  -37.1681466  -8.253602370 -10.74569227
## [51,]  3.404767e+01   14.7805080  -34.2781083  -2.710003818  -3.81858991
## [52,]  1.478051e+01  279.3493471  265.5788160 -18.628910244 -27.32638158
## [53,] -3.427811e+01  265.5788160  497.8925313   2.825912636   1.27288976
## [54,] -2.710004e+00  -18.6289102    2.8259126   6.184179902   7.21771652
## [55,] -3.818590e+00  -27.3263816    1.2728898   7.217716521  12.04920643
## [56,] -3.387775e+00  -23.3413334    3.2774008   7.434037293   9.28653604
## [57,] -3.310877e+00  -24.2106341    1.7689908   6.917002710  10.78812022
## [58,] -3.552540e+00  -24.0193255    0.8394858   6.875839027  10.07344441
## [59,] -3.536076e+00  -22.8069265    5.9313971   7.271618802  10.55602052
## [60,] -1.289556e+00  -36.4169349  -22.3417630   6.877726452  10.08039313
## [61,] -4.724998e+00    3.8016025   54.4252135   7.458746141  10.74613073
## [62,]  1.279855e+01   -8.2792649   19.2649562   7.301683898  10.37001537
## [63,] -1.574764e+01   -4.6418685   16.9767404   3.073738095   3.72543442
## [64,] -7.752887e+00 -130.3543176 -117.2962414  15.880627316  26.37637189
## [65,]  1.306448e+01 -114.5683458 -218.6209573  -0.342523366  -0.24732806
## [66,]  5.152604e-04    0.1073713    0.1410330  -0.007704525  -0.01501070
##              [,56]        [,57]        [,58]        [,59]        [,60]
##  [1,]   1.41816259   0.47194748   0.49731662   1.16179657   0.54605355
##  [2,]  -2.04659087  -1.96463690  -1.81237082  -2.23850756  -1.77412149
##  [3,]  -1.28680037   0.37502225   0.15608641  -0.60895831   0.06451224
##  [4,]  -0.69165835  -1.12107162  -0.50307344  -1.48789673  -0.65335634
##  [5,]  -1.68421831  -0.04159788  -0.57823077  -0.94300406  -0.55825062
##  [6,]  -1.15051869  -0.88201606  -0.21202838  -1.55251322  -0.30780131
##  [7,]  -1.64427677  -0.17096908  -0.71969692  -0.63540258  -1.65064195
##  [8,]  -1.34476571  -0.62676673  -0.45588965  -1.54453455   0.96869667
##  [9,]  -1.47484588  -0.41150534  -0.50940787  -1.00430209  -1.30345447
## [10,]  -1.38510429  -0.48663010  -0.57661444  -1.08226578  -1.01300300
## [11,]  -1.07384898  -0.77407691  -0.71539593  -1.06201997  -0.65030565
## [12,]  -2.24244126   0.31865814   0.12457280  -1.39653851   0.05390037
## [13,]  -1.06691791  -1.09662118  -1.01080658  -1.23934748  -0.88542587
## [14,] -17.65896008 -17.24354182 -16.69881366 -18.14878073 -16.53206037
## [15,]  16.89279680  18.19262266  17.61385234  17.86061009  17.43807316
## [16,]   9.68261285   9.67606187   9.29612984  10.12011358   9.18145516
## [17,]  -9.33549131  -9.60313145  -9.36227386  -9.65513665  -9.30080060
## [18,]  16.57029992  16.43940460  15.79635096  17.25448716  15.69084274
## [19,]  19.20429186  17.50640916  17.50137664  18.64726000  17.18035313
## [20,]  15.39458018  17.58541663  15.93697859  17.97954933  16.02797748
## [21,]  18.59821536  17.03092336  17.11107772  18.11436480  16.77022501
## [22,]  16.71531820  17.79182045  15.52970328  18.92979575  16.25276897
## [23,]  18.40289474  16.76295745  17.48587162  17.76060033  16.87585446
## [24,]  17.51609880  17.23212924  16.43748560  17.77148216  19.64085210
## [25,]  17.71306545  17.31902664  16.93895550  18.62402530  12.68457350
## [26,]  17.42415163  17.10979946  16.25369501  17.88170622  18.13801297
## [27,]   9.63353439   9.20255270   8.92674965  10.20349330   5.71522915
## [28,]  37.63654240  37.01886729  36.07102720  38.26136752  38.64626988
## [29,]   4.33759582   4.58486160   4.64151573   3.62628210  11.90769142
## [30,] -13.86248280 -12.28928557 -12.28660276 -13.23317151 -12.37403621
## [31,] -15.96032150 -21.48378194 -19.56761871 -19.89748634 -19.43898511
## [32,] -21.91579111 -15.58328679 -18.54306999 -16.59273901 -17.49491845
## [33,] -15.13720636 -19.64195064 -16.74795839 -18.56036137 -17.19062619
## [34,] -18.49845603 -16.59272641 -19.62347972 -16.04363672 -18.03120679
## [35,] -15.80529142 -19.27040297 -16.16014656 -19.73100880 -15.58561759
## [36,] -17.36764896 -17.54205147 -18.06090610 -16.36986628 -23.17923483
## [37,] -16.42430945 -18.47205203 -17.33714850 -18.77539835 -11.82691876
## [38,] -17.66063051 -18.52920415 -17.48852302 -19.34734586 -12.03015066
## [39,]  -8.30059494  -8.34590616  -8.34901197  -8.05572155 -10.93736459
## [40,] -36.95677240 -41.74671070 -40.05200990 -40.25636496 -38.38859504
## [41,]  -5.47873306  -4.33828599  -4.25579278  -5.38016426  -0.61256812
## [42,]  -8.88485302  -9.42411585  -9.02155689  -9.52597304  -8.86435149
## [43,] -10.52397703  -9.90989818  -9.73045866 -10.53202048  -9.56861279
## [44,]  -8.07264290 -10.06612478  -9.11760535  -9.88398954  -9.03618100
## [45,] -10.31298431  -9.45773671  -9.43200391 -10.18494753  -9.24460439
## [46,]  -8.98515014 -10.27710776  -8.17965358 -10.97695295  -9.10217609
## [47,] -10.25562736  -9.15935425 -10.07479968  -9.66369376  -9.04569186
## [48,]  -9.82747788  -9.55222573  -9.15172176  -9.49384520 -14.30867796
## [49,]  -9.09713615  -9.69256548  -9.47041523 -10.56827714  -2.05689860
## [50,] -10.04342339  -9.86551384  -8.63199491 -11.09231534  -7.38999356
## [51,]  -3.38777504  -3.31087707  -3.55253996  -3.53607617  -1.28955550
## [52,] -23.34133335 -24.21063411 -24.01932548 -22.80692648 -36.41693495
## [53,]   3.27740084   1.76899081   0.83948577   5.93139711 -22.34176298
## [54,]   7.43403729   6.91700271   6.87583903   7.27161880   6.87772645
## [55,]   9.28653604  10.78812022  10.07344441  10.55602052  10.08039313
## [56,]  11.02960526   8.74461657   9.92355907   9.00804206   9.39482256
## [57,]   8.74461657  10.07029976   8.91788245  10.04702235   9.18906193
## [58,]   9.92355907   8.91788245  10.56291638   8.51754023   9.52463956
## [59,]   9.00804206  10.04702235   8.51754023  10.72919713   8.46852791
## [60,]   9.39482256   9.18906193   9.52463956   8.46852791  14.55557208
## [61,]   9.60030052  10.07168260   9.35974516  11.39418915   0.41987753
## [62,]   9.57231892   9.70430768   9.04386776  10.23614341   7.85191823
## [63,]   3.73407652   3.54257625   3.67582262   3.68728486   2.76609613
## [64,]  22.05940521  23.84615763  23.49104562  22.41614535  29.74673954
## [65,]  -0.71119123  -0.32049650   0.29598294  -2.35312164  12.01643547
## [66,]  -0.01074083  -0.01319809  -0.01284438  -0.01115969  -0.02022568
##               [,61]         [,62]         [,63]         [,64]         [,65]
##  [1,]   1.401289228   1.212602611   0.319914612    0.29809834   -2.81484480
##  [2,]  -2.520419376  -2.254587118  -0.419470620   -4.84798201    2.52869894
##  [3,]  -0.823815993  -0.712822266  -0.272238519    2.15128985    3.04188180
##  [4,]  -1.666557529  -1.299923097  -0.256600385   -1.72530839    2.49702648
##  [5,]  -1.181535456  -1.174918957  -0.341064949    0.47290254    2.95916485
##  [6,]  -1.900867552  -1.186052187  -0.402568011   -0.81545429    2.86630741
##  [7,]   0.200249725  -1.637684163   0.065721902   -0.25037921    2.00355840
##  [8,]  -3.525252460  -0.699621661  -0.812664296    0.01083985    4.06334571
##  [9,]   0.003007229  -1.825943322  -0.030800667    0.18509724    3.30650860
## [10,]  -0.039161317   1.786676775  -1.613027858   -3.18508068   -1.11195255
## [11,]  -1.504353976  -2.264499204   0.780978343   -0.10516287    3.96268662
## [12,]  -1.992369248  -1.824272918  -0.616090353    1.57692849    2.11742050
## [13,]  -1.426641588  -0.909378614  -0.713770334   -7.53850868   -6.26563601
## [14,] -19.159548130 -18.207035389  -6.142816014  -41.40845344    6.26894689
## [15,]  18.286486173  17.800052881   6.395903494   45.25087282   -0.13246860
## [16,]  10.726807818  10.133151315   3.189701408   23.66963397   -3.98133925
## [17,]  -9.906123692  -9.650125316  -3.600338250  -23.35732514    0.44505280
## [18,]  18.137245691  17.228331247   5.776541513   39.60158092   -5.70642567
## [19,]  19.858432455  18.999433496   6.446135812   42.15618815   -6.91622457
## [20,]  18.686043193  17.485110324   5.866050613   41.88688202   -5.16120685
## [21,]  19.326217718  18.552533431   6.239157801   41.05773107   -6.92653000
## [22,]  19.330020524  18.509871367   5.920385594   41.48777587   -6.39025396
## [23,]  18.753768875  17.905451809   6.369987989   41.21344736   -6.04203296
## [24,]  14.306163749  20.670367420   4.395381659   41.74528764   -4.24864848
## [25,]  24.999706654  13.862246337   8.974698229   41.51261134   -8.42479385
## [26,]  17.668046164  27.302079454   0.052559636   40.15133887   -3.33178523
## [27,]  16.047903001   6.885576344  10.025298941   14.53999312  -20.97536085
## [28,]  33.956448942  35.644760514  11.603709908  103.25504640   11.93592559
## [29,] -10.854813661   1.230237294  -5.665697955   40.06753197   53.01062448
## [30,] -13.522390129 -13.280353336  -5.837799736  -27.55136078    0.53210172
## [31,] -20.186564425 -19.314413071  -6.481187155  -54.23857210   -1.03246120
## [32,] -17.757840154 -17.974510748  -6.903215814  -40.02617965    1.92395552
## [33,] -18.517315095 -17.584875386  -6.173462556  -47.63719173   -1.05197250
## [34,] -17.402313470 -17.587292066  -6.439171365  -43.74039611    0.55907492
## [35,] -20.764096803 -17.469031929  -6.810370219  -45.73811714    0.89062663
## [36,]  -9.341645762 -17.674925869  -5.242226384  -48.84247717   -8.05601492
## [37,] -29.764459802 -16.885114544  -7.363669162  -44.12224088    3.01203601
## [38,] -32.605243914 -38.084016874   1.398571476  -23.36305241   33.19070271
## [39,]  -2.652013969   0.183211382  -9.140935116  -31.04289719  -16.86347636
## [40,] -43.713165639 -37.786493637 -15.962004787  -90.13795065   33.71819053
## [41,] -13.548721974  -7.356391618  -4.225229248   29.97960376   66.43530900
## [42,] -10.057979865  -9.530513730  -2.771825596  -23.91665029    3.25250821
## [43,] -11.226757986 -10.632591828  -3.437563279  -24.08808875    4.38062976
## [44,] -10.369147576  -9.683570912  -2.930112229  -24.76985729    3.04513415
## [45,] -10.890231552 -10.303942406  -3.318016365  -23.14995837    4.42579576
## [46,] -10.927029730 -10.904544029  -2.874991943  -23.47736109    4.85290218
## [47,] -10.766676593  -9.079590935  -3.681572265  -23.96406516    2.89523955
## [48,]  -3.085776297 -14.694870707  -0.364810255  -23.08892734    2.68047793
## [49,] -20.386634212   2.933942152  -9.220427828  -31.84292055   -5.94825989
## [50,] -20.189434963 -38.688348531  11.241063272   -5.58951829   24.87242055
## [51,]  -4.724997936  12.798548720 -15.747638623   -7.75288686   13.06448278
## [52,]   3.801602460  -8.279264875  -4.641868478 -130.35431757 -114.56834584
## [53,]  54.425213461  19.264956227  16.976740354 -117.29624144 -218.62095732
## [54,]   7.458746141   7.301683898   3.073738095   15.88062732   -0.34252337
## [55,]  10.746130731  10.370015373   3.725434425   26.37637189   -0.24732806
## [56,]   9.600300520   9.572318924   3.734076522   22.05940521   -0.71119123
## [57,]  10.071682605   9.704307683   3.542576252   23.84615763   -0.32049650
## [58,]   9.359745156   9.043867762   3.675822617   23.49104562    0.29598294
## [59,]  11.394189145  10.236143414   3.687284864   22.41614535   -2.35312164
## [60,]   0.419877525   7.851918225   2.766096128   29.74673954   12.01643547
## [61,]  28.856487627  15.679540578   4.131275654    8.70453479  -27.18055237
## [62,]  15.679540578  25.213130799  -4.395961998   13.43567727  -11.89216295
## [63,]   4.131275654  -4.395961998   9.158757080    7.71406441   -5.72098084
## [64,]   8.704534786  13.435677267   7.714064412   92.75382044   55.10536190
## [65,] -27.180552368 -11.892162950  -5.720980841   55.10536190  101.79573889
## [66,]   0.005026522  -0.001164111  -0.002851076   -0.07115196   -0.06573649
##               [,66]
##  [1,]  0.0083629520
##  [2,] -0.0036912960
##  [3,] -0.0110243865
##  [4,] -0.0064235420
##  [5,] -0.0094405203
##  [6,] -0.0077187226
##  [7,] -0.0084653387
##  [8,] -0.0087450000
##  [9,] -0.0090373689
## [10,] -0.0048693702
## [11,] -0.0052797415
## [12,] -0.0185129830
## [13,]  0.0048222221
## [14,]  0.0151311160
## [15,] -0.0293585011
## [16,] -0.0093083596
## [17,]  0.0124941464
## [18,] -0.0140353307
## [19,] -0.0147448825
## [20,] -0.0168445534
## [21,] -0.0143086780
## [22,] -0.0154685527
## [23,] -0.0146141579
## [24,] -0.0156729167
## [25,] -0.0150974647
## [26,] -0.0140850427
## [27,]  0.0022090058
## [28,] -0.0505711941
## [29,] -0.0376871689
## [30,]  0.0153761865
## [31,]  0.0382678509
## [32,]  0.0219780855
## [33,]  0.0329622673
## [34,]  0.0274614874
## [35,]  0.0299601132
## [36,]  0.0336520993
## [37,]  0.0284441242
## [38,]  0.0033576025
## [39,]  0.0249936847
## [40,]  0.0509653152
## [41,] -0.0415782443
## [42,]  0.0103315667
## [43,]  0.0089979631
## [44,]  0.0114450272
## [45,]  0.0083240224
## [46,]  0.0092210831
## [47,]  0.0094303426
## [48,]  0.0087353181
## [49,]  0.0190192928
## [50,] -0.0112765326
## [51,]  0.0005152604
## [52,]  0.1073712613
## [53,]  0.1410330161
## [54,] -0.0077045249
## [55,] -0.0150106984
## [56,] -0.0107408290
## [57,] -0.0131980859
## [58,] -0.0128443826
## [59,] -0.0111596882
## [60,] -0.0202256793
## [61,]  0.0050265216
## [62,] -0.0011641106
## [63,] -0.0028510761
## [64,] -0.0711519636
## [65,] -0.0657364872
## [66,]  0.0050831669
## 
## $log_evidence
## [1] -367.8151
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 225

3d)

A function is defined for you in the code chunk below. This function creates a coefficient summary plot in the style of the coefplot() function, but uses the Bayesian results from the Laplace Approximation. The first argument is the vector of posterior means, and the second argument is the vector of posterior standard deviations. The third argument is the name of the feature associated with each coefficient.

viz_post_coefs <- function(post_means, post_sds, xnames)
{
  tibble::tibble(
    mu = post_means,
    sd = post_sds,
    x = xnames
  ) %>% 
    mutate(x = factor(x, levels = xnames)) %>% 
    ggplot(mapping = aes(x = x)) +
    geom_hline(yintercept = 0, color = 'grey', linetype = 'dashed') +
    geom_point(mapping = aes(y = mu)) +
    geom_linerange(mapping = aes(ymin = mu - 2 * sd,
                                 ymax = mu + 2 * sd,
                                 group = x)) +
    labs(x = 'feature', y = 'coefficient value') +
    coord_flip() +
    theme_bw()
}

Create the posterior summary visualization figure for model 3 and model 6. You must provide the posterior means and standard deviations of the regression coefficients (the \(\beta\) parameters). Do NOT include the \(\varphi\) parameter. The feature names associated with the coefficients can be extracted from the design matrix using the colnames() function.

SOLUTION

### make the posterior coefficient visualization for model 3
post_means_3 <- laplace_03_weak$mode[1:ncol(X03)]
post_sds_3 <- sqrt(diag(laplace_03_weak$var_matrix)[1:ncol(X03)])
xnames_3 <- colnames(X03)

viz_post_coefs(post_means_3, post_sds_3, xnames_3)

### make the posterior coefficient visualization for model 6
post_means_6 <- laplace_06_weak$mode[1:ncol(X06)]
post_sds_6 <- sqrt(diag(laplace_06_weak$var_matrix)[1:ncol(X06)])
xnames_6 <- colnames(X06)

viz_post_coefs(post_means_6, post_sds_6, xnames_6)

3e)

Use the Bayes Factor to identify the better of the models.

SOLUTION

bf <- exp(laplace_03_weak$log_evidence - laplace_06_weak$log_evidence)
bf

## [1] 1.483395e+88

bf is greater than 1, then Model 3 is favored over Model 6.

3f)

You fit the Bayesian models assuming a diffuse or weak prior. Let’s now try a more informative or strong prior by reducing the prior standard deviation on the regression coefficients from 50 to 1. The prior mean will still be zero.

Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.

Run the Laplace Approximation using the strong prior for both the model 3 and model 6 formulations. Assign the results to laplace_03_strong and laplace_06_strong.

Confirm that the optimizations converged for both laplace approximation results.

SOLUTION

Define the lists of required information for the strong prior.

info_03_strong <- list(
  yobs = df$y,
  design_matrix = X03 ,
  mu_beta = 0,
  tau_beta = 1,
  sigma_rate = 1 
)

info_06_strong <- list(
  yobs = df$y,
  design_matrix = X06 ,
  mu_beta = 0,
  tau_beta = 1,
  sigma_rate = 1
)

Execute the Laplace Approximation.

laplace_03_strong <- my_laplace(rep(0, ncol(X03)+1), lm_logpost, info_03_strong)
laplace_03_strong

## $mode
##  [1]  0.651611994  0.162911105 -0.152720109 -0.051739764 -0.548518942
##  [6]  0.118438289 -0.079147677  0.002641276  0.017129670 -0.398729134
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  0.0135459465 -2.873342e-04 -6.802239e-03  3.977466e-04 -5.671510e-03
##  [2,] -0.0002873342  7.716870e-03  1.696316e-04  7.895995e-04 -5.465928e-04
##  [3,] -0.0068022386  1.696316e-04  7.455471e-03  1.562535e-04  3.314479e-03
##  [4,]  0.0003977466  7.895995e-04  1.562535e-04  8.086040e-03  5.740609e-04
##  [5,] -0.0056715099 -5.465928e-04  3.314479e-03  5.740609e-04  5.582223e-03
##  [6,]  0.0023593363 -3.371891e-05  1.560380e-04  5.950124e-04 -1.881045e-03
##  [7,] -0.0016154431 -2.928137e-03  5.844256e-04 -1.063982e-03  1.486907e-03
##  [8,]  0.0012803512  1.764089e-04 -5.707081e-04 -2.435573e-03 -1.561454e-03
##  [9,]  0.0024572993  3.145148e-04 -3.430504e-03 -8.954330e-04 -2.355648e-03
## [10,] -0.0001338042 -1.539843e-05  7.467039e-05  2.359958e-06  7.772145e-05
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,]  2.359336e-03 -1.615443e-03  1.280351e-03  0.0024572993 -1.338042e-04
##  [2,] -3.371891e-05 -2.928137e-03  1.764089e-04  0.0003145148 -1.539843e-05
##  [3,]  1.560380e-04  5.844256e-04 -5.707081e-04 -0.0034305043  7.467039e-05
##  [4,]  5.950124e-04 -1.063982e-03 -2.435573e-03 -0.0008954330  2.359958e-06
##  [5,] -1.881045e-03  1.486907e-03 -1.561454e-03 -0.0023556476  7.772145e-05
##  [6,]  1.197120e-02 -4.426886e-03  4.166809e-03 -0.0027336429 -4.238226e-05
##  [7,] -4.426886e-03  4.913270e-03 -2.503061e-03  0.0008204165  3.291125e-05
##  [8,]  4.166809e-03 -2.503061e-03  4.411329e-03 -0.0004354008 -2.628054e-05
##  [9,] -2.733643e-03  8.204165e-04 -4.354008e-04  0.0034170232 -3.183560e-05
## [10,] -4.238226e-05  3.291125e-05 -2.628054e-05 -0.0000318356  5.001155e-03
## 
## $log_evidence
## [1] -130.0251
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 68

laplace_06_strong <- my_laplace(rep(0, ncol(X06)+1), lm_logpost, info_06_strong)
laplace_06_strong

## $mode
##  [1]  0.3211913265 -0.0287158412 -0.1003958688  0.0565332450  0.7264568887
##  [6]  0.6243006390  0.1763818239  0.2158223802  0.3251240416 -0.2592733804
## [11]  0.8798073185  0.1410245934 -0.0375778966 -0.2760562036 -0.5764863280
## [16] -0.0787398641 -0.1936144200  0.4182706557 -0.1789016722 -0.1459875878
## [21]  0.3162221037  0.4518055172  0.7714941420 -0.4720480845 -0.6949870776
## [26]  0.9824389708 -0.0210462673 -0.0924572965  0.2747596768  0.4446096263
## [31] -0.1195110240  0.4518271183  0.1670136520  0.1369458446 -0.2047773966
## [36] -0.2346348028  0.1132497656  0.0004949989 -0.0504843236  0.3037346453
## [41] -0.0703279982 -0.1454908139  0.3849427236 -0.0871220868  0.0072681023
## [46] -0.3653196899  0.2061070095  0.6151077657 -0.0794043413  0.4087221759
## [51]  0.0048028222  0.0791329340  0.1492803814  0.1128845746  0.1629053375
## [56]  0.5346662507 -0.1515009892  0.2565331633  0.0343110363  0.1988994227
## [61] -0.0789994282  0.0737712445  0.1599361434  0.3559950287 -0.0149655836
## [66] -0.5995244271
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  7.388174e-02 -0.0600678247 -0.0700382386 -6.318613e-02 -0.0692512918
##  [2,] -6.006782e-02  0.3030062755 -0.0522858236  9.295783e-02  0.0421114524
##  [3,] -7.003824e-02 -0.0522858236  0.2342879775 -1.080196e-02  0.0966652412
##  [4,] -6.318613e-02  0.0929578323 -0.0108019608  2.575234e-01 -0.0326681871
##  [5,] -6.925129e-02  0.0421114524  0.0966652412 -3.266819e-02  0.2391796638
##  [6,] -6.188423e-02  0.0601916798  0.0460859119  9.917829e-02 -0.0256945347
##  [7,] -5.925631e-02  0.0471794638  0.0624238873  3.886402e-02  0.0824027514
##  [8,] -6.645314e-02  0.0579105664  0.0632415345  6.225274e-02  0.0576271254
##  [9,] -6.680026e-02  0.0549304527  0.0646731549  5.578318e-02  0.0664243770
## [10,] -6.275190e-02  0.0543955399  0.0610999064  5.587470e-02  0.0599828837
## [11,] -4.630010e-02  0.0781844267  0.0243306781  4.552251e-02  0.0412840081
## [12,] -1.202121e-01  0.0020223579  0.1647911304  8.962013e-02  0.1235546515
## [13,] -1.888208e-03  0.0606082705 -0.0249695420  1.286208e-02 -0.0015117671
## [14,] -1.550701e-03  0.0124369972  0.0009073611 -4.000300e-03  0.0041238719
## [15,] -2.782302e-02  0.0098215946  0.0149967780  1.459757e-02  0.0124782008
## [16,]  4.094782e-03 -0.0049918044 -0.0052256147 -2.691460e-05 -0.0057924597
## [17,] -1.195132e-03  0.0109292299  0.0048038525  6.512298e-03  0.0074202749
## [18,]  9.086353e-03 -0.0153095776  0.0108005984 -7.698314e-03 -0.0095012875
## [19,] -2.827570e-03  0.0005821239  0.0298062122 -2.971546e-02  0.0200128508
## [20,] -3.680405e-03 -0.0029752701 -0.0109439354 -2.754006e-02 -0.0114304446
## [21,]  9.907856e-04 -0.0172679936  0.0128424562 -2.229178e-02  0.0247057473
## [22,]  1.124633e-03 -0.0084112852 -0.0034434692  1.035710e-02 -0.0095617203
## [23,] -2.035957e-05 -0.0077620540  0.0024672814  2.489333e-03 -0.0001258530
## [24,]  2.638150e-03 -0.0102166487 -0.0035818801  6.224495e-03 -0.0108062677
## [25,]  1.424759e-03 -0.0095137904 -0.0019121183  9.613606e-04 -0.0040549482
## [26,]  2.767374e-03 -0.0089495550 -0.0002779611  1.352037e-03 -0.0015891195
## [27,]  1.155339e-02 -0.0197242645 -0.0013997207 -9.570321e-03 -0.0089019749
## [28,] -1.114971e-02  0.0229452158 -0.0043472585  2.410561e-02  0.0091316632
## [29,]  3.494102e-03 -0.0153777841  0.0094422225 -5.884072e-03  0.0002999189
## [30,]  4.593746e-03 -0.1619182966  0.0367044287 -9.230592e-03  0.0090286865
## [31,]  4.485356e-03  0.0230950510 -0.1014355663 -1.408428e-02  0.0067101644
## [32,]  2.938015e-03  0.0160786508 -0.0048567418 -1.012579e-01  0.0035767866
## [33,]  1.044547e-02 -0.0002614818 -0.0132942451  3.704924e-02 -0.1437512786
## [34,]  1.164670e-02 -0.0027002549 -0.0060281909 -2.681018e-03  0.0177471322
## [35,]  9.633549e-03 -0.0018549160  0.0016255552 -1.128571e-02  0.0106737783
## [36,]  8.234626e-03 -0.0012177722 -0.0031513396 -4.631090e-04 -0.0040585766
## [37,]  1.093352e-03  0.0009840262  0.0015898495 -4.686235e-04  0.0029220016
## [38,]  1.082618e-02 -0.0031616727 -0.0038850611 -3.792334e-03 -0.0033757556
## [39,]  1.431654e-02 -0.0065697857 -0.0125576231 -8.660753e-03 -0.0096654898
## [40,] -8.426945e-03  0.0037437271  0.0209073368  1.107747e-02  0.0153558989
## [41,] -7.156315e-03  0.0068305006  0.0026216197  6.339574e-03  0.0053641608
## [42,]  4.902059e-03 -0.0037605748  0.0069187760 -1.887681e-03 -0.0025442542
## [43,] -1.008877e-02  0.0014168834 -0.0060747527  6.190288e-03  0.0058905711
## [44,] -2.399782e-03 -0.0064290815  0.0126925609  1.846283e-02 -0.0027626719
## [45,] -7.665168e-04  0.0078584040 -0.0002285053  1.404000e-03  0.0090978271
## [46,] -5.395759e-03  0.0050030377  0.0029050765  7.664360e-03 -0.0042367623
## [47,] -2.030077e-03  0.0017628199  0.0060189298 -6.714512e-03  0.0142907578
## [48,] -9.233957e-03  0.0052731139  0.0090204842  3.281035e-03  0.0098012439
## [49,] -3.520565e-03  0.0010156825  0.0032134074 -4.241638e-04  0.0031108232
## [50,] -6.783709e-03 -0.0014608471  0.0047945220  2.922073e-03  0.0028337160
## [51,] -2.362744e-04 -0.0034572446 -0.0029227583  5.607287e-06 -0.0036541563
## [52,] -2.767269e-03 -0.0029092004  0.0084405398 -3.785341e-03  0.0055741682
## [53,]  3.821890e-03 -0.0039456232 -0.0052048283 -3.500127e-04 -0.0046631325
## [54,]  4.294682e-03 -0.0025503525 -0.0050289737 -8.162137e-03 -0.0087722711
## [55,]  6.442085e-03 -0.0060660404  0.0049222658 -1.789517e-03 -0.0181082767
## [56,]  1.337724e-02 -0.0239183300  0.0015896458 -5.216399e-03  0.0119448206
## [57,]  3.003910e-03 -0.0115227081 -0.0084600767 -8.146446e-03  0.0070699698
## [58,]  8.684649e-03 -0.0119222741 -0.0061551110 -1.740828e-02  0.0017583326
## [59,]  2.801285e-03 -0.0126097990 -0.0084595432 -2.572132e-03 -0.0231338260
## [60,]  9.763095e-03 -0.0143444079 -0.0097048353 -1.275592e-02 -0.0101778198
## [61,]  1.040441e-02 -0.0115929036 -0.0094977385 -1.184057e-02 -0.0101525183
## [62,]  5.728068e-03 -0.0093743772 -0.0065714029 -9.370131e-03 -0.0078864502
## [63,] -3.419157e-03 -0.0089431018  0.0062415693 -1.684691e-03  0.0019293485
## [64,]  2.096081e-02 -0.0085647920 -0.0323573398 -2.398484e-02 -0.0266088419
## [65,]  3.527435e-04 -0.0113280199  0.0067236546 -2.681663e-03  0.0012888016
## [66,]  1.158441e-03 -0.0001973227 -0.0015274813 -4.290852e-04 -0.0019518854
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,] -0.0618842321 -5.925631e-02 -0.0664531423 -0.0668002646 -0.0627518986
##  [2,]  0.0601916798  4.717946e-02  0.0579105664  0.0549304527  0.0543955399
##  [3,]  0.0460859119  6.242389e-02  0.0632415345  0.0646731549  0.0610999064
##  [4,]  0.0991782890  3.886402e-02  0.0622527420  0.0557831833  0.0558746984
##  [5,] -0.0256945347  8.240275e-02  0.0576271254  0.0664243770  0.0599828837
##  [6,]  0.2843377225 -2.680813e-02  0.0792397798  0.0485565785  0.0599658903
##  [7,] -0.0268081288  3.358085e-01 -0.0424115212  0.0935442053  0.0335976958
##  [8,]  0.0792397798 -4.241152e-02  0.2434517478 -0.0090292666  0.0925853516
##  [9,]  0.0485565785  9.354421e-02 -0.0090292666  0.2401879106 -0.0343087938
## [10,]  0.0599658903  3.359770e-02  0.0925853516 -0.0343087938  0.3010910887
## [11,]  0.0377474448  4.536802e-02  0.0266350269  0.0825079210 -0.0698403672
## [12,]  0.1023344255  1.045179e-01  0.1124277668  0.1162889489  0.0937935544
## [13,]  0.0054664106 -2.318981e-03  0.0107854171 -0.0178495926  0.0540534890
## [14,]  0.0053376648 -6.205929e-03  0.0108384797  0.0031806419  0.0089943057
## [15,]  0.0071112144  6.727900e-03  0.0050704655  0.0177401713  0.0024878190
## [16,] -0.0062791430  6.896358e-05 -0.0093112527 -0.0046721594 -0.0082701636
## [17,]  0.0084639520  8.482367e-03  0.0107492027  0.0044300486  0.0112036817
## [18,] -0.0111980958 -1.557830e-03 -0.0158543022 -0.0093686763 -0.0143552807
## [19,] -0.0120827662  1.158475e-02 -0.0085831051  0.0009280840 -0.0056279831
## [20,]  0.0057561899  4.738992e-03 -0.0020090119  0.0011862383 -0.0023777743
## [21,] -0.0105554312  8.035368e-03 -0.0127842187 -0.0009330341 -0.0078870767
## [22,]  0.0166622178 -9.152818e-03 -0.0005198756 -0.0046243046 -0.0062779131
## [23,]  0.0029359553 -2.477954e-02 -0.0165025412  0.0052106196 -0.0010536704
## [24,]  0.0053277025 -5.140940e-02  0.0730001585 -0.0152753226 -0.0125673969
## [25,] -0.0053437907  3.341541e-03 -0.0114209495  0.0311337997 -0.0125742214
## [26,] -0.0027149406  8.328857e-03 -0.0087220381 -0.0193559707  0.0987045231
## [27,] -0.0081743965 -9.088509e-03 -0.0032848920 -0.0162971511 -0.0088093268
## [28,]  0.0101570022  2.207109e-02  0.0052139897  0.0134734571 -0.0094601074
## [29,] -0.0021562917  2.678789e-03 -0.0047348759  0.0085627754 -0.0280918682
## [30,]  0.0029174405  6.943371e-03  0.0049699297  0.0001272717  0.0063952332
## [31,] -0.0037198462 -3.634633e-03  0.0060217631 -0.0012707472  0.0052388076
## [32,]  0.0042796508 -1.327938e-03  0.0058417926 -0.0004915936  0.0048045661
## [33,]  0.0215288163  2.370570e-03 -0.0039569092 -0.0019917259  0.0008124596
## [34,] -0.1384295313 -7.192888e-03  0.0196552425 -0.0135817665  0.0074332853
## [35,]  0.0024039577 -1.595212e-01  0.0311269218 -0.0093346513  0.0053276825
## [36,]  0.0031155289  2.193179e-02 -0.1574421779  0.0170328586  0.0001579182
## [37,] -0.0025222754  1.802122e-02 -0.0040405059 -0.1203509017  0.0137246341
## [38,]  0.0010052004 -2.454028e-03 -0.0036662901  0.0135641746 -0.1169013460
## [39,] -0.0088996486 -1.557485e-03 -0.0153605413  0.0074078858 -0.0240251969
## [40,]  0.0141422688  1.972688e-02  0.0122580207  0.0197112591  0.0012930903
## [41,]  0.0041508928  5.443640e-03  0.0049529797  0.0070004742 -0.0083791767
## [42,]  0.0023588900 -3.818215e-03  0.0048398745 -0.0023487173  0.0045659435
## [43,]  0.0107923213  3.118076e-03  0.0112549928  0.0091369241  0.0100106551
## [44,]  0.0029369298 -1.384114e-02  0.0108639596  0.0016205267  0.0057058968
## [45,]  0.0088585200  5.642291e-03  0.0061587659  0.0026979428  0.0074821252
## [46,] -0.0169018413  1.376471e-02  0.0111645001  0.0030303035  0.0073407268
## [47,]  0.0080227826  1.436263e-02  0.0016512116  0.0049870147  0.0029089920
## [48,]  0.0041290069  2.250044e-02 -0.0201317686 -0.0100194254  0.0162104633
## [49,]  0.0032943612 -1.633059e-03 -0.0186458855  0.0272998353 -0.0149281571
## [50,]  0.0024706185  1.563780e-03 -0.0016969538  0.0042296092  0.0032232805
## [51,] -0.0025860339 -5.436069e-03 -0.0014379579 -0.0028543327 -0.0231743717
## [52,]  0.0045709272 -9.244758e-04  0.0084096630  0.0038817021 -0.0031715684
## [53,] -0.0041736178  4.468101e-04 -0.0074914239 -0.0012976842 -0.0097534001
## [54,] -0.0096883269 -8.844557e-03 -0.0114954197 -0.0066478285 -0.0115796191
## [55,] -0.0075187482 -1.221456e-02 -0.0127565064 -0.0083240307 -0.0131202684
## [56,] -0.0201079600  1.903089e-03 -0.0183788508 -0.0119987824 -0.0140776200
## [57,] -0.0055884903 -2.028691e-02 -0.0077163074 -0.0073197534 -0.0113459869
## [58,] -0.0049292833  1.853078e-02 -0.0221427761 -0.0072825735 -0.0147321550
## [59,]  0.0118005917 -7.632556e-03 -0.0057542892 -0.0089844422 -0.0089625452
## [60,] -0.0148935874 -8.274553e-03  0.0203523546  0.0051728290 -0.0162957306
## [61,] -0.0110782960 -1.280079e-02  0.0151152760 -0.0149840042 -0.0028193023
## [62,] -0.0078359324 -9.636796e-03 -0.0086588268 -0.0013665130 -0.0074097083
## [63,]  0.0000881407 -3.332741e-04  0.0021410759 -0.0049519681  0.0182957365
## [64,] -0.0233372402 -2.830553e-02 -0.0263821304 -0.0254267385 -0.0111549692
## [65,]  0.0024200813 -3.141391e-03  0.0025387744 -0.0009411105  0.0163543524
## [66,] -0.0017369352 -1.938593e-03 -0.0005927517 -0.0016449567 -0.0003278450
##               [,11]        [,12]         [,13]         [,14]         [,15]
##  [1,] -4.630010e-02 -0.120212105 -1.888208e-03 -0.0015507012 -0.0278230167
##  [2,]  7.818443e-02  0.002022358  6.060827e-02  0.0124369972  0.0098215946
##  [3,]  2.433068e-02  0.164791130 -2.496954e-02  0.0009073611  0.0149967780
##  [4,]  4.552251e-02  0.089620132  1.286208e-02 -0.0040003001  0.0145975651
##  [5,]  4.128401e-02  0.123554652 -1.511767e-03  0.0041238719  0.0124782008
##  [6,]  3.774744e-02  0.102334426  5.466411e-03  0.0053376648  0.0071112144
##  [7,]  4.536802e-02  0.104517899 -2.318981e-03 -0.0062059292  0.0067279003
##  [8,]  2.663503e-02  0.112427767  1.078542e-02  0.0108384797  0.0050704655
##  [9,]  8.250792e-02  0.116288949 -1.784959e-02  0.0031806419  0.0177401713
## [10,] -6.984037e-02  0.093793554  5.405349e-02  0.0089943057  0.0024878190
## [11,]  2.416619e-01  0.013250525 -8.640185e-02  0.0043422160  0.0219562579
## [12,]  1.325052e-02  0.450198107  5.776840e-02 -0.0067400832  0.0046553810
## [13,] -8.640185e-02  0.057768397  3.824925e-01 -0.0002533215 -0.0074295101
## [14,]  4.342216e-03 -0.006740083 -2.533215e-04  0.0859640886 -0.0082749776
## [15,]  2.195626e-02  0.004655381 -7.429510e-03 -0.0082749776  0.1267406308
## [16,] -2.242863e-03 -0.006367182 -1.990767e-03 -0.0293026218  0.0113353104
## [17,] -8.718037e-04  0.020088214  3.200534e-03  0.0113530073 -0.0486897302
## [18,] -1.389449e-02  0.008574372 -9.761102e-03 -0.0723826903  0.0014942587
## [19,]  3.291219e-03 -0.003098193  5.676872e-03 -0.0637733794  0.0209356928
## [20,] -1.132256e-03  0.012779736 -1.206851e-03 -0.0558131593  0.0116898527
## [21,] -3.893305e-03  0.008095030 -1.226140e-03 -0.0733337969  0.0077738231
## [22,] -2.826239e-03  0.004869188  4.895345e-05 -0.0646529070  0.0049323286
## [23,] -4.030865e-03  0.012084257  1.944733e-03 -0.0596173387 -0.0079445861
## [24,] -2.544774e-03  0.003513290 -1.233151e-03 -0.0688708586  0.0051272267
## [25,] -5.847191e-03  0.003100646  6.419829e-03 -0.0558245913  0.0157939853
## [26,] -3.400507e-02 -0.007986949 -2.962640e-03 -0.0556023669 -0.0122767656
## [27,] -3.098056e-02  0.018985631  3.613324e-02 -0.0184609335 -0.0195814608
## [28,]  3.212559e-02 -0.003024520 -2.186449e-02 -0.1007318925 -0.0206001452
## [29,]  3.744990e-02 -0.025204091 -1.108691e-01  0.0030965408 -0.0155605922
## [30,] -5.394854e-03  0.015553422  1.294222e-03  0.0055631356 -0.0608853627
## [31,] -2.994614e-03  0.002717528  6.280245e-03  0.0128117540 -0.0477657954
## [32,]  1.047671e-03 -0.001101974  7.304802e-03  0.0074103651 -0.0402092121
## [33,] -8.593447e-03  0.006152289  3.132242e-03  0.0072944847 -0.0678389568
## [34,] -1.328031e-02  0.008578176  5.115637e-03  0.0023922016 -0.0788817554
## [35,] -9.418109e-03  0.011657290  5.331985e-03 -0.0006088929 -0.0774128550
## [36,] -7.860600e-03  0.005379019  4.408843e-03 -0.0036521661 -0.0548534911
## [37,]  8.009089e-05  0.005644747  6.421357e-04  0.0035081909 -0.0228267628
## [38,] -7.213859e-03  0.008584889  9.105655e-03 -0.0019546805 -0.0672228464
## [39,] -1.172716e-01  0.027128033  2.998261e-02 -0.0017470598 -0.0379103653
## [40,]  3.602594e-02 -0.112521356 -5.914511e-03 -0.0247461147 -0.0615969155
## [41,]  2.566126e-02 -0.004501804 -1.973709e-01 -0.0036783861  0.0131163041
## [42,] -1.381262e-02  0.022512507 -8.496759e-03  0.0229422006 -0.0377263264
## [43,]  8.458107e-03  0.006553870  3.295950e-03  0.0246423117  0.0073630150
## [44,] -1.840346e-03  0.010767684 -1.168503e-03  0.0148525847 -0.0103746417
## [45,]  9.140833e-04  0.002129326  4.586386e-03  0.0278850284 -0.0240583016
## [46,]  2.016064e-03  0.008019441  1.777548e-03  0.0146160333 -0.0027477220
## [47,]  2.256809e-03  0.002129469  2.138501e-03  0.0180895855 -0.0072691282
## [48,]  4.275278e-03  0.008960051 -6.985036e-04  0.0136578509  0.0073137285
## [49,]  1.045169e-02  0.006850435  4.017028e-03  0.0098286963  0.0078062103
## [50,]  6.626123e-03 -0.003787212 -1.423081e-02 -0.0139567170  0.0268346569
## [51,]  1.789227e-03 -0.001091437  1.484803e-02 -0.0144450887  0.0205642605
## [52,]  6.548378e-03  0.008358445 -2.077289e-02  0.0137339646 -0.0112653098
## [53,]  1.664028e-02 -0.027009221 -4.007051e-02 -0.0203988331 -0.0024845039
## [54,] -6.009452e-03 -0.010353641 -7.555466e-03 -0.0146403860  0.0357974833
## [55,] -1.745415e-03 -0.026325630 -9.902701e-04 -0.0102225216  0.0324782154
## [56,] -9.588607e-03 -0.021963620 -2.010303e-03 -0.0114688173  0.0018715350
## [57,] -9.126114e-04 -0.021347937 -2.891586e-03 -0.0116540697  0.0440417747
## [58,] -2.966724e-03 -0.026424031 -2.421527e-04 -0.0136186344  0.0221776199
## [59,] -1.437684e-03 -0.019370601 -3.812035e-03 -0.0007215787  0.0415662736
## [60,] -7.346019e-03 -0.025968068 -3.332772e-03 -0.0059461596  0.0155788496
## [61,] -1.694378e-03 -0.014597599  6.412798e-03  0.0043901875  0.0003724539
## [62,]  5.255333e-03 -0.016302550 -1.735602e-03  0.0072478160  0.0176508293
## [63,]  1.595621e-02 -0.011284697 -1.017475e-02  0.0052923990  0.0190135027
## [64,] -2.098268e-02 -0.030288776  9.938115e-03  0.0045922966  0.0347588717
## [65,] -1.222436e-02  0.013875413  5.171887e-02  0.0104690932 -0.0104445371
## [66,] -2.822354e-03 -0.001529455  1.338728e-03  0.0009941847  0.0006513935
##               [,16]         [,17]        [,18]         [,19]         [,20]
##  [1,]  4.094782e-03 -0.0011951320  0.009086353 -0.0028275701 -3.680405e-03
##  [2,] -4.991804e-03  0.0109292299 -0.015309578  0.0005821239 -2.975270e-03
##  [3,] -5.225615e-03  0.0048038525  0.010800598  0.0298062122 -1.094394e-02
##  [4,] -2.691460e-05  0.0065122984 -0.007698314 -0.0297154609 -2.754006e-02
##  [5,] -5.792460e-03  0.0074202749 -0.009501287  0.0200128508 -1.143044e-02
##  [6,] -6.279143e-03  0.0084639520 -0.011198096 -0.0120827662  5.756190e-03
##  [7,]  6.896358e-05  0.0084823673 -0.001557830  0.0115847521  4.738992e-03
##  [8,] -9.311253e-03  0.0107492027 -0.015854302 -0.0085831051 -2.009012e-03
##  [9,] -4.672159e-03  0.0044300486 -0.009368676  0.0009280840  1.186238e-03
## [10,] -8.270164e-03  0.0112036817 -0.014355281 -0.0056279831 -2.377774e-03
## [11,] -2.242863e-03 -0.0008718037 -0.013894494  0.0032912195 -1.132256e-03
## [12,] -6.367182e-03  0.0200882138  0.008574372 -0.0030981933  1.277974e-02
## [13,] -1.990767e-03  0.0032005338 -0.009761102  0.0056768725 -1.206851e-03
## [14,] -2.930262e-02  0.0113530073 -0.072382690 -0.0637733794 -5.581316e-02
## [15,]  1.133531e-02 -0.0486897302  0.001494259  0.0209356928  1.168985e-02
## [16,]  5.891221e-02  0.0170854219  0.023482779  0.0157191923  1.498618e-02
## [17,]  1.708542e-02  0.0512068666 -0.014436508 -0.0178858429 -1.195679e-02
## [18,]  2.348278e-02 -0.0144365076  0.480181004 -0.0585125473  6.542216e-02
## [19,]  1.571919e-02 -0.0178858429 -0.058512547  0.4844080291 -9.009648e-02
## [20,]  1.498618e-02 -0.0119567917  0.065422163 -0.0900964799  5.626614e-01
## [21,]  2.156565e-02 -0.0121956775  0.060240730  0.0824860422 -6.463277e-02
## [22,]  1.584611e-02 -0.0112438228  0.056023191  0.0402282436  6.954319e-02
## [23,]  1.465323e-02 -0.0038869686  0.052641499  0.0412472492  3.334743e-02
## [24,]  1.954385e-02 -0.0100303757  0.057213437  0.0496040384  4.939204e-02
## [25,]  1.495279e-02 -0.0142744162  0.045340962  0.0447529726  3.631935e-02
## [26,]  4.908235e-03 -0.0051026564  0.049609404  0.0395007927  3.608538e-02
## [27,]  3.704836e-03  0.0068874954  0.072043312 -0.0024751030  1.299357e-02
## [28,]  9.392049e-03 -0.0104149603 -0.053708209  0.1099006161  6.136106e-02
## [29,]  3.358936e-03  0.0099570799  0.079613244 -0.0259508334 -4.125468e-04
## [30,] -1.917023e-02  0.0157670309 -0.038591427  0.0122647916 -2.865601e-02
## [31,] -1.524091e-02  0.0131051615 -0.011510792  0.1078442634  1.628665e-04
## [32,] -1.465103e-02  0.0097744916 -0.021147647  0.0138836179  4.617744e-02
## [33,] -1.295221e-02  0.0215030862  0.001736177 -0.0304483019 -1.132455e-02
## [34,] -6.601048e-03  0.0288207595  0.005230445 -0.0220583287 -6.516430e-03
## [35,] -6.223703e-03  0.0289386748  0.002113749 -0.0035056143 -4.358827e-03
## [36,] -1.079727e-03  0.0187974085  0.004487630 -0.0025084057  2.158262e-03
## [37,] -4.485426e-03  0.0049007326 -0.000862003 -0.0050592640 -3.170161e-03
## [38,] -6.637053e-03  0.0215371637  0.005572699 -0.0061992115 -1.484772e-03
## [39,]  4.027293e-03  0.0180536258  0.004284863  0.0004062804 -1.872899e-03
## [40,] -2.458978e-02 -0.0035174353  0.022331037  0.0050197315  1.762944e-02
## [41,]  2.394066e-03  0.0011179853  0.002857199  0.0101627476  2.407180e-03
## [42,] -3.928317e-02  0.0042849727 -0.188175599 -0.0129189541 -7.419199e-03
## [43,] -6.105475e-02 -0.0271050894  0.017824991 -0.1057689068  3.504479e-02
## [44,] -4.937853e-02 -0.0172295521 -0.006616627  0.0061485308 -1.990570e-01
## [45,] -5.836631e-02 -0.0110432953 -0.026451958 -0.0094477197  2.684578e-02
## [46,] -4.653007e-02 -0.0176756024 -0.010844617 -0.0119473014  4.605268e-03
## [47,] -5.358822e-02 -0.0200813334 -0.017514535  0.0055305877 -2.195500e-02
## [48,] -5.162043e-02 -0.0258769180 -0.010034121 -0.0024342443 -3.774976e-03
## [49,] -3.559688e-02 -0.0211593528 -0.007697566 -0.0004587154 -3.969198e-03
## [50,] -2.825887e-02 -0.0340256375  0.013270253  0.0182068438  1.385489e-02
## [51,] -6.045598e-04 -0.0258180468 -0.010795949  0.0171428330  1.173162e-02
## [52,] -8.375842e-02 -0.0218082657  0.048488043 -0.0059466407 -3.704134e-03
## [53,]  2.049146e-02 -0.0104305405 -0.013801658  0.0161600877  1.268248e-02
## [54,] -1.020895e-02 -0.0374298216 -0.021110350  0.0075661163  1.966409e-02
## [55,] -1.795295e-02 -0.0457777897  0.025338258 -0.0089003024  1.686015e-02
## [56,] -1.491381e-02 -0.0287687695  0.016577960  0.0220284103 -3.156180e-02
## [57,] -1.680979e-02 -0.0495189257  0.016294360  0.0130501036  2.071868e-02
## [58,] -1.390964e-02 -0.0372088617  0.009952941  0.0341604678  1.951720e-02
## [59,] -2.495776e-02 -0.0501925430  0.011715849 -0.0049922760  1.361234e-03
## [60,] -2.232410e-02 -0.0379917766  0.011260732  0.0106828093  6.223554e-03
## [61,] -2.694200e-02 -0.0275585569  0.001557646  0.0006178148 -5.271324e-04
## [62,] -3.115579e-02 -0.0376624019 -0.001210974  0.0017050247 -1.098122e-03
## [63,] -2.384871e-02 -0.0224721667 -0.008640609 -0.0023998052 -4.518773e-06
## [64,] -5.142726e-02 -0.0852791175  0.024885820  0.0185187425  2.249966e-03
## [65,] -3.254530e-02 -0.0057663702 -0.016942081 -0.0104712607 -3.865148e-03
## [66,]  1.170427e-03  0.0007823946 -0.003350941  0.0004903678 -1.845255e-04
##               [,21]         [,22]         [,23]         [,24]         [,25]
##  [1,]  0.0009907856  1.124633e-03 -2.035957e-05  0.0026381503  0.0014247592
##  [2,] -0.0172679936 -8.411285e-03 -7.762054e-03 -0.0102166487 -0.0095137904
##  [3,]  0.0128424562 -3.443469e-03  2.467281e-03 -0.0035818801 -0.0019121183
##  [4,] -0.0222917817  1.035710e-02  2.489333e-03  0.0062244946  0.0009613606
##  [5,]  0.0247057473 -9.561720e-03 -1.258530e-04 -0.0108062677 -0.0040549482
##  [6,] -0.0105554312  1.666222e-02  2.935955e-03  0.0053277025 -0.0053437907
##  [7,]  0.0080353681 -9.152818e-03 -2.477954e-02 -0.0514093951  0.0033415414
##  [8,] -0.0127842187 -5.198756e-04 -1.650254e-02  0.0730001585 -0.0114209495
##  [9,] -0.0009330341 -4.624305e-03  5.210620e-03 -0.0152753226  0.0311337997
## [10,] -0.0078870767 -6.277913e-03 -1.053670e-03 -0.0125673969 -0.0125742214
## [11,] -0.0038933048 -2.826239e-03 -4.030865e-03 -0.0025447742 -0.0058471907
## [12,]  0.0080950303  4.869188e-03  1.208426e-02  0.0035132898  0.0031006464
## [13,] -0.0012261404  4.895345e-05  1.944733e-03 -0.0012331507  0.0064198293
## [14,] -0.0733337969 -6.465291e-02 -5.961734e-02 -0.0688708586 -0.0558245913
## [15,]  0.0077738231  4.932329e-03 -7.944586e-03  0.0051272267  0.0157939853
## [16,]  0.0215656546  1.584611e-02  1.465323e-02  0.0195438537  0.0149527911
## [17,] -0.0121956775 -1.124382e-02 -3.886969e-03 -0.0100303757 -0.0142744162
## [18,]  0.0602407299  5.602319e-02  5.264150e-02  0.0572134374  0.0453409624
## [19,]  0.0824860422  4.022824e-02  4.124725e-02  0.0496040384  0.0447529726
## [20,] -0.0646327738  6.954319e-02  3.334743e-02  0.0493920363  0.0363193454
## [21,]  0.3985571004 -7.230001e-02  8.053273e-02  0.0439778406  0.0515360732
## [22,] -0.0723000074  4.814415e-01 -8.584290e-02  0.0732966952  0.0361673503
## [23,]  0.0805327263 -8.584290e-02  4.941713e-01 -0.0582452134  0.0689991915
## [24,]  0.0439778406  7.329670e-02 -5.824521e-02  0.4048233331 -0.0618114577
## [25,]  0.0515360732  3.616735e-02  6.899919e-02 -0.0618114577  0.4876343961
## [26,]  0.0479595615  4.708846e-02  3.269253e-02  0.0597297155 -0.0732910320
## [27,]  0.0159769239  1.250437e-02  1.988358e-02  0.0091630264  0.0526227697
## [28,]  0.0871876561  7.778600e-02  7.689093e-02  0.0809297547  0.0796577844
## [29,] -0.0024463305 -2.387865e-03 -4.613509e-04 -0.0024938603 -0.0093604826
## [30,]  0.0016888559 -3.303895e-03  4.028646e-03 -0.0052880002 -0.0094125152
## [31,] -0.0101830899 -8.449330e-03  2.852326e-03 -0.0080119326 -0.0108520621
## [32,] -0.0222853652 -7.446415e-03  3.933325e-03 -0.0047237127 -0.0088407907
## [33,]  0.0103195822  1.435094e-02  1.362744e-02 -0.0069730809 -0.0106290526
## [34,] -0.0009931488  1.453565e-02 -1.947479e-02  0.0130951406 -0.0084623828
## [35,]  0.0042627497 -2.543908e-02 -2.686771e-02 -0.0039974653 -0.0052469471
## [36,] -0.0037025730  1.963284e-03 -2.724693e-02  0.0068827341  0.0086981338
## [37,] -0.0029485427 -5.134531e-04 -1.138932e-02 -0.0049068814  0.0757105992
## [38,]  0.0012262941  3.225658e-03  1.244347e-02  0.0017671286 -0.0029278292
## [39,]  0.0003825325  9.223576e-04  1.405693e-03  0.0045085646 -0.0056235981
## [40,]  0.0235025147  2.293774e-02  2.740832e-02  0.0217173079  0.0129604221
## [41,]  0.0031804235  1.904098e-03  2.249320e-03  0.0023583189  0.0082140283
## [42,] -0.0203085896 -1.237928e-02 -9.964897e-03 -0.0139907556 -0.0136775373
## [43,] -0.0224165809 -1.151140e-02 -1.288155e-02 -0.0167458254 -0.0103490331
## [44,]  0.0144632492 -6.016985e-03  6.602718e-03 -0.0032192523 -0.0069999094
## [45,] -0.1161064054  2.394903e-02 -2.986026e-02 -0.0198362859 -0.0146233312
## [46,] -0.0246728176 -2.075274e-01  5.568316e-02  0.0165328522 -0.0125805671
## [47,]  0.0299041400  4.182429e-02 -2.160146e-01  0.0041047761 -0.0096219537
## [48,] -0.0089357141 -8.080836e-03  2.013336e-03 -0.1933540471  0.0349096728
## [49,] -0.0014176109 -1.289925e-02  3.877026e-02  0.0272876984 -0.1724487185
## [50,]  0.0136584744  1.669819e-02  4.506044e-03  0.0170933206  0.0073659712
## [51,]  0.0128682642  1.056324e-02  1.133392e-02  0.0072910347  0.0412338129
## [52,] -0.0054391931 -9.103331e-04  1.046768e-03 -0.0059436468  0.0071491864
## [53,]  0.0164749288  1.398663e-02  1.228453e-02  0.0153821656  0.0066937745
## [54,]  0.0129775794  1.331842e-02  6.685237e-03  0.0131984257  0.0152347157
## [55,]  0.0091310321  1.123347e-02  5.313420e-03  0.0091113030  0.0120282075
## [56,]  0.0219066903  8.841205e-03 -2.892289e-03  0.0090327713  0.0104892576
## [57,] -0.0021542478  1.372011e-02  6.851414e-03  0.0111463202  0.0145355131
## [58,] -0.0065757252 -4.321209e-02  1.207838e-02  0.0191879069  0.0095008979
## [59,]  0.0093395539  6.564501e-02  3.978848e-02 -0.0006985245  0.0107797611
## [60,]  0.0095024519  2.585545e-03  3.146281e-02  0.0372460444 -0.0125690922
## [61,] -0.0021158524  1.479359e-03 -1.982642e-02 -0.0089147907  0.0305692711
## [62,] -0.0033458448 -2.520089e-03 -3.438155e-03 -0.0057297335  0.0120085659
## [63,] -0.0021543553 -1.026163e-03 -4.365230e-03 -0.0016012053 -0.0139470074
## [64,]  0.0007927347  3.639094e-03 -5.865609e-03  0.0006104786 -0.0033836824
## [65,] -0.0076411454 -3.111429e-03 -5.038739e-03 -0.0043066024 -0.0116242270
## [66,] -0.0020663603 -2.290041e-03 -3.596957e-03  0.0017221013  0.0024711107
##               [,26]         [,27]         [,28]         [,29]         [,30]
##  [1,]  0.0027673742  0.0115533910 -0.0111497072  0.0034941016  0.0045937458
##  [2,] -0.0089495550 -0.0197242645  0.0229452158 -0.0153777841 -0.1619182966
##  [3,] -0.0002779611 -0.0013997207 -0.0043472585  0.0094422225  0.0367044287
##  [4,]  0.0013520371 -0.0095703213  0.0241056135 -0.0058840722 -0.0092305918
##  [5,] -0.0015891195 -0.0089019749  0.0091316632  0.0002999189  0.0090286865
##  [6,] -0.0027149406 -0.0081743965  0.0101570022 -0.0021562917  0.0029174405
##  [7,]  0.0083288566 -0.0090885088  0.0220710929  0.0026787888  0.0069433707
##  [8,] -0.0087220381 -0.0032848920  0.0052139897 -0.0047348759  0.0049699297
##  [9,] -0.0193559707 -0.0162971511  0.0134734571  0.0085627754  0.0001272717
## [10,]  0.0987045231 -0.0088093268 -0.0094601074 -0.0280918682  0.0063952332
## [11,] -0.0340050710 -0.0309805640  0.0321255918  0.0374498972 -0.0053948540
## [12,] -0.0079869495  0.0189856310 -0.0030245197 -0.0252040906  0.0155534218
## [13,] -0.0029626398  0.0361332391 -0.0218644873 -0.1108691388  0.0012942225
## [14,] -0.0556023669 -0.0184609335 -0.1007318925  0.0030965408  0.0055631356
## [15,] -0.0122767656 -0.0195814608 -0.0206001452 -0.0155605922 -0.0608853627
## [16,]  0.0049082348  0.0037048364  0.0093920492  0.0033589359 -0.0191702274
## [17,] -0.0051026564  0.0068874954 -0.0104149603  0.0099570799  0.0157670309
## [18,]  0.0496094040  0.0720433116 -0.0537082088  0.0796132437 -0.0385914268
## [19,]  0.0395007927 -0.0024751030  0.1099006161 -0.0259508334  0.0122647916
## [20,]  0.0360853775  0.0129935658  0.0613610584 -0.0004125468 -0.0286560089
## [21,]  0.0479595615  0.0159769239  0.0871876561 -0.0024463305  0.0016888559
## [22,]  0.0470884645  0.0125043665  0.0777860046 -0.0023878649 -0.0033038950
## [23,]  0.0326925274  0.0198835776  0.0768909281 -0.0004613509  0.0040286455
## [24,]  0.0597297155  0.0091630264  0.0809297547 -0.0024938603 -0.0052880002
## [25,] -0.0732910320  0.0526227697  0.0796577844 -0.0093604826 -0.0094125152
## [26,]  0.5980027229 -0.1487519180  0.0124021025  0.0102313954  0.0112841069
## [27,] -0.1487519180  0.5382017204  0.0350982133 -0.0517673562 -0.0044704012
## [28,]  0.0124021025  0.0350982133  0.6082581194  0.0505175496  0.0420197973
## [29,]  0.0102313954 -0.0517673562  0.0505175496  0.5641650697 -0.0085175352
## [30,]  0.0112841069 -0.0044704012  0.0420197973 -0.0085175352  0.6906921937
## [31,] -0.0017489915  0.0045198316  0.0031584389  0.0036600539 -0.0414641356
## [32,]  0.0051159394  0.0008411301  0.0122665800  0.0009755946  0.0355438112
## [33,]  0.0060371345  0.0096917854  0.0103025049  0.0088827417  0.0209056209
## [34,]  0.0082583224  0.0130351037  0.0160752077  0.0089668076  0.0311246254
## [35,]  0.0103987444  0.0139348242  0.0235671169  0.0097591476  0.0396415388
## [36,]  0.0022365789  0.0117907055  0.0195347584  0.0049537801  0.0270034586
## [37,] -0.0002276930  0.0014974287  0.0062570799  0.0082399602  0.0124481667
## [38,]  0.0449844652 -0.0358105154 -0.0033909017 -0.0004049058  0.0345797344
## [39,] -0.0388382714 -0.0911689952  0.0075890933  0.0479822649  0.0453725715
## [40,]  0.0074691652  0.0033283251 -0.0109485467 -0.0517418813 -0.0319464189
## [41,] -0.0116178887  0.0281650620 -0.0588829307 -0.1644545433  0.0360487396
## [42,] -0.0040663301 -0.0145383873  0.0391820595 -0.0243602304 -0.1611306876
## [43,] -0.0023064948 -0.0025472251 -0.0149110721 -0.0006268913  0.0723154726
## [44,]  0.0028604159  0.0008563228  0.0025090579 -0.0017684343  0.0060363454
## [45,] -0.0023132216 -0.0020198597 -0.0031365303 -0.0030199864  0.0328502067
## [46,]  0.0022899921 -0.0021512780  0.0025695888 -0.0046690845  0.0094760142
## [47,]  0.0002724940 -0.0010534224  0.0041811948 -0.0038088680  0.0228732917
## [48,]  0.0245682123 -0.0160033611 -0.0010642582 -0.0061556046  0.0116456946
## [49,] -0.0331357086  0.0278007265  0.0208301705  0.0061747921  0.0054867057
## [50,] -0.2028869534 -0.0780762790  0.0041676316  0.0092415104 -0.0026148193
## [51,] -0.0639558675 -0.1954138897  0.0284947390  0.0287180974 -0.0101130528
## [52,] -0.0070587637 -0.0030374178 -0.1437866164  0.0183109324  0.0297966455
## [53,]  0.0195491747  0.0028111971  0.0298810586 -0.1587036296 -0.0031878315
## [54,]  0.0069614386 -0.0008741449  0.0104048649 -0.0053938553 -0.1639789470
## [55,]  0.0067163111 -0.0044091636  0.0118247514 -0.0075883607 -0.0030360204
## [56,]  0.0103639422  0.0018058184  0.0180973595 -0.0032965339  0.0136201360
## [57,]  0.0054380730 -0.0055586039  0.0111261284 -0.0092681686 -0.0169344433
## [58,]  0.0107580603 -0.0035121117  0.0196120035 -0.0077917067  0.0102696303
## [59,] -0.0014299856 -0.0055745424 -0.0003620087 -0.0073863110 -0.0225637508
## [60,]  0.0111751585 -0.0118017857  0.0036731127 -0.0130438428  0.0007975820
## [61,] -0.0130678251  0.0293778669  0.0284116775  0.0161002331  0.0063786523
## [62,]  0.0318297449  0.0575277358  0.0157409355 -0.0217203074  0.0001250982
## [63,]  0.0582839880  0.0515255017  0.0037412315 -0.0203014894 -0.0162726597
## [64,]  0.0384247868 -0.0030166177  0.0263098612 -0.0169109756  0.0355385629
## [65,]  0.0251602559 -0.0111205966 -0.0001738525  0.0243445610 -0.0082050842
## [66,] -0.0062966488  0.0022280846 -0.0006468791 -0.0020580772 -0.0041365886
##               [,31]         [,32]         [,33]         [,34]         [,35]
##  [1,]  0.0044853558  0.0029380153  1.044547e-02  0.0116466967  0.0096335492
##  [2,]  0.0230950510  0.0160786508 -2.614818e-04 -0.0027002549 -0.0018549160
##  [3,] -0.1014355663 -0.0048567418 -1.329425e-02 -0.0060281909  0.0016255552
##  [4,] -0.0140842772 -0.1012578599  3.704924e-02 -0.0026810183 -0.0112857102
##  [5,]  0.0067101644  0.0035767866 -1.437513e-01  0.0177471322  0.0106737783
##  [6,] -0.0037198462  0.0042796508  2.152882e-02 -0.1384295313  0.0024039577
##  [7,] -0.0036346329 -0.0013279379  2.370570e-03 -0.0071928879 -0.1595211925
##  [8,]  0.0060217631  0.0058417926 -3.956909e-03  0.0196552425  0.0311269218
##  [9,] -0.0012707472 -0.0004915936 -1.991726e-03 -0.0135817665 -0.0093346513
## [10,]  0.0052388076  0.0048045661  8.124596e-04  0.0074332853  0.0053276825
## [11,] -0.0029946137  0.0010476705 -8.593447e-03 -0.0132803122 -0.0094181090
## [12,]  0.0027175277 -0.0011019742  6.152289e-03  0.0085781759  0.0116572902
## [13,]  0.0062802453  0.0073048015  3.132242e-03  0.0051156366  0.0053319852
## [14,]  0.0128117540  0.0074103651  7.294485e-03  0.0023922016 -0.0006088929
## [15,] -0.0477657954 -0.0402092121 -6.783896e-02 -0.0788817554 -0.0774128550
## [16,] -0.0152409130 -0.0146510256 -1.295221e-02 -0.0066010485 -0.0062237028
## [17,]  0.0131051615  0.0097744916  2.150309e-02  0.0288207595  0.0289386748
## [18,] -0.0115107922 -0.0211476471  1.736177e-03  0.0052304452  0.0021137489
## [19,]  0.1078442634  0.0138836179 -3.044830e-02 -0.0220583287 -0.0035056143
## [20,]  0.0001628665  0.0461774411 -1.132455e-02 -0.0065164297 -0.0043588265
## [21,] -0.0101830899 -0.0222853652  1.031958e-02 -0.0009931488  0.0042627497
## [22,] -0.0084493296 -0.0074464149  1.435094e-02  0.0145356496 -0.0254390777
## [23,]  0.0028523257  0.0039333248  1.362744e-02 -0.0194747871 -0.0268677147
## [24,] -0.0080119326 -0.0047237127 -6.973081e-03  0.0130951406 -0.0039974653
## [25,] -0.0108520621 -0.0088407907 -1.062905e-02 -0.0084623828 -0.0052469471
## [26,] -0.0017489915  0.0051159394  6.037134e-03  0.0082583224  0.0103987444
## [27,]  0.0045198316  0.0008411301  9.691785e-03  0.0130351037  0.0139348242
## [28,]  0.0031584389  0.0122665800  1.030250e-02  0.0160752077  0.0235671169
## [29,]  0.0036600539  0.0009755946  8.882742e-03  0.0089668076  0.0097591476
## [30,] -0.0414641356  0.0355438112  2.090562e-02  0.0311246254  0.0396415388
## [31,]  0.7066038077 -0.0816296942  6.195210e-02  0.0495237818  0.0268668300
## [32,] -0.0816296942  0.7411227112 -2.839544e-02  0.0479617490  0.0190534923
## [33,]  0.0619521025 -0.0283954407  5.884981e-01 -0.1015384458  0.0761694899
## [34,]  0.0495237818  0.0479617490 -1.015384e-01  0.6023001486 -0.0929277852
## [35,]  0.0268668300  0.0190534923  7.616949e-02 -0.0929277852  0.5725737930
## [36,]  0.0167813196  0.0162055485  2.034672e-02  0.0402636448 -0.0596553258
## [37,]  0.0109084789  0.0085915660  1.651588e-02  0.0105089959  0.0248370459
## [38,]  0.0251236150  0.0225762354  3.515861e-02  0.0454029999  0.0348759245
## [39,]  0.0139753349  0.0089584699  2.017776e-02  0.0223047686  0.0230232331
## [40,]  0.0252053578  0.0285060995  3.782349e-02  0.0415821516  0.0444258301
## [41,] -0.0037278812 -0.0058303273 -6.471458e-03 -0.0087249303 -0.0060194816
## [42,] -0.0333533923  0.0175478879  2.787713e-02  0.0255080630  0.0219887804
## [43,]  0.0574846357  0.0387729789  3.999757e-03 -0.0088778947 -0.0044334134
## [44,]  0.0397264926 -0.1000291693  6.025124e-02  0.0582120943 -0.0045916047
## [45,] -0.0002094054  0.0371089727 -5.414571e-02 -0.0091270431  0.0164777512
## [46,]  0.0134744514  0.0146169145  1.517315e-02 -0.0044497593  0.0221301988
## [47,] -0.0027339545  0.0059869348 -3.066635e-02  0.0433687569  0.0337406398
## [48,]  0.0054146579  0.0087012249  2.841949e-03 -0.0119020243 -0.0037179782
## [49,]  0.0032136408  0.0040152188 -1.301777e-03 -0.0042483553 -0.0150233843
## [50,] -0.0057108023 -0.0015654409 -1.067431e-02 -0.0140427890 -0.0188420296
## [51,] -0.0108466575 -0.0054220985 -9.145367e-03 -0.0099459176 -0.0113952865
## [52,]  0.0285635859  0.0179419961  1.355682e-02  0.0052046923  0.0066780226
## [53,] -0.0082864502 -0.0008506838  1.810981e-03  0.0036596443  0.0041946220
## [54,] -0.0013752159 -0.0151563802 -1.561357e-02 -0.0204977955 -0.0211957510
## [55,] -0.1724986504  0.0391261259 -7.191055e-03 -0.0177997091 -0.0202140588
## [56,] -0.0040413081 -0.2427381226 -5.343302e-02 -0.0369794743  0.0096030124
## [57,] -0.0019875815  0.0204712934 -1.302130e-01  0.0344121973 -0.0426987724
## [58,] -0.0451500487 -0.0180597694 -3.157139e-02 -0.1937800008  0.0435562481
## [59,]  0.0227605217  0.0027094843  3.468068e-02  0.0158360680 -0.2003814410
## [60,] -0.0010816742  0.0007174240 -4.129133e-03 -0.0092968329  0.0162173107
## [61,]  0.0074810063  0.0057913246  7.480836e-03 -0.0041563878  0.0176569382
## [62,]  0.0012658839  0.0014271605 -3.785482e-03 -0.0105087253 -0.0102770638
## [63,] -0.0067505679 -0.0006721685 -6.847785e-03 -0.0129389634 -0.0094132852
## [64,]  0.0114439363  0.0035841644 -6.748054e-03 -0.0181538508 -0.0229148853
## [65,]  0.0023271543  0.0084661544  6.995568e-03  0.0074793219  0.0018986146
## [66,]  0.0012970194 -0.0026372534 -8.377401e-05  0.0002258945  0.0007066990
##               [,36]         [,37]         [,38]         [,39]        [,40]
##  [1,]  0.0082346259  1.093352e-03  0.0108261821  1.431654e-02 -0.008426945
##  [2,] -0.0012177722  9.840262e-04 -0.0031616727 -6.569786e-03  0.003743727
##  [3,] -0.0031513396  1.589849e-03 -0.0038850611 -1.255762e-02  0.020907337
##  [4,] -0.0004631090 -4.686235e-04 -0.0037923339 -8.660753e-03  0.011077474
##  [5,] -0.0040585766  2.922002e-03 -0.0033757556 -9.665490e-03  0.015355899
##  [6,]  0.0031155289 -2.522275e-03  0.0010052004 -8.899649e-03  0.014142269
##  [7,]  0.0219317933  1.802122e-02 -0.0024540278 -1.557485e-03  0.019726879
##  [8,] -0.1574421779 -4.040506e-03 -0.0036662901 -1.536054e-02  0.012258021
##  [9,]  0.0170328586 -1.203509e-01  0.0135641746  7.407886e-03  0.019711259
## [10,]  0.0001579182  1.372463e-02 -0.1169013460 -2.402520e-02  0.001293090
## [11,] -0.0078606003  8.009089e-05 -0.0072138594 -1.172716e-01  0.036025938
## [12,]  0.0053790193  5.644747e-03  0.0085848889  2.712803e-02 -0.112521356
## [13,]  0.0044088426  6.421357e-04  0.0091056553  2.998261e-02 -0.005914511
## [14,] -0.0036521661  3.508191e-03 -0.0019546805 -1.747060e-03 -0.024746115
## [15,] -0.0548534911 -2.282676e-02 -0.0672228464 -3.791037e-02 -0.061596916
## [16,] -0.0010797275 -4.485426e-03 -0.0066370533  4.027293e-03 -0.024589776
## [17,]  0.0187974085  4.900733e-03  0.0215371637  1.805363e-02 -0.003517435
## [18,]  0.0044876299 -8.620030e-04  0.0055726992  4.284863e-03  0.022331037
## [19,] -0.0025084057 -5.059264e-03 -0.0061992115  4.062804e-04  0.005019732
## [20,]  0.0021582619 -3.170161e-03 -0.0014847718 -1.872899e-03  0.017629435
## [21,] -0.0037025730 -2.948543e-03  0.0012262941  3.825325e-04  0.023502515
## [22,]  0.0019632839 -5.134531e-04  0.0032256581  9.223576e-04  0.022937742
## [23,] -0.0272469315 -1.138932e-02  0.0124434683  1.405693e-03  0.027408322
## [24,]  0.0068827341 -4.906881e-03  0.0017671286  4.508565e-03  0.021717308
## [25,]  0.0086981338  7.571060e-02 -0.0029278292 -5.623598e-03  0.012960422
## [26,]  0.0022365789 -2.276930e-04  0.0449844652 -3.883827e-02  0.007469165
## [27,]  0.0117907055  1.497429e-03 -0.0358105154 -9.116900e-02  0.003328325
## [28,]  0.0195347584  6.257080e-03 -0.0033909017  7.589093e-03 -0.010948547
## [29,]  0.0049537801  8.239960e-03 -0.0004049058  4.798226e-02 -0.051741881
## [30,]  0.0270034586  1.244817e-02  0.0345797344  4.537257e-02 -0.031946419
## [31,]  0.0167813196  1.090848e-02  0.0251236150  1.397533e-02  0.025205358
## [32,]  0.0162055485  8.591566e-03  0.0225762354  8.958470e-03  0.028506100
## [33,]  0.0203467178  1.651588e-02  0.0351586117  2.017776e-02  0.037823492
## [34,]  0.0402636448  1.050900e-02  0.0454029999  2.230477e-02  0.041582152
## [35,] -0.0596553258  2.483705e-02  0.0348759245  2.302323e-02  0.044425830
## [36,]  0.6181032134 -3.485777e-02  0.0341874991  1.622305e-02  0.031738323
## [37,] -0.0348577720  7.835516e-01 -0.0503545770  1.772848e-02  0.023497756
## [38,]  0.0341874991 -5.035458e-02  0.7069703637 -1.642916e-01 -0.025377397
## [39,]  0.0162230539  1.772848e-02 -0.1642915565  6.875218e-01 -0.002923179
## [40,]  0.0317383229  2.349776e-02 -0.0253773968 -2.923179e-03  0.815607293
## [41,] -0.0062772000  8.765050e-04  0.0093037095  4.370735e-03  0.031578987
## [42,]  0.0137416280  7.660978e-03  0.0191864843  3.723719e-03  0.035166048
## [43,] -0.0069247166  1.341357e-03 -0.0023981827 -1.041435e-02  0.020414938
## [44,] -0.0043478509  5.078483e-03  0.0061066166 -2.620141e-03  0.025988008
## [45,]  0.0110540776  5.528948e-03  0.0139532031 -1.288390e-04  0.030327932
## [46,]  0.0209728381  1.659857e-03  0.0030050540 -4.218527e-03  0.020373240
## [47,] -0.0064661729  7.864083e-03  0.0010368248 -4.223771e-03  0.026530813
## [48,]  0.0853669241  2.946874e-02 -0.0043972661 -5.787776e-03  0.020783920
## [49,]  0.0251281592  5.823881e-02 -0.0410972349 -1.487145e-02  0.022176957
## [50,] -0.0124885235  1.105960e-02  0.0365069633  5.913660e-02  0.035949045
## [51,] -0.0030458401 -1.221582e-02  0.0329518578 -2.355858e-02  0.009704179
## [52,] -0.0008410007 -3.687256e-04  0.0205737817 -1.472512e-02 -0.003822303
## [53,]  0.0056693212  7.027481e-03 -0.0002725252  4.858314e-05 -0.062333437
## [54,] -0.0134370155 -4.279515e-03 -0.0160153759 -2.871788e-02  0.042041074
## [55,] -0.0123022908 -1.878563e-03 -0.0128855055 -1.116040e-02  0.006322271
## [56,]  0.0092020572  1.517526e-03  0.0034578773 -3.437439e-03  0.023088148
## [57,] -0.0194340532 -4.387914e-03 -0.0191297657 -1.652242e-02  0.005076973
## [58,]  0.0110086988 -1.233000e-03 -0.0072106187 -9.112227e-03  0.014349410
## [59,]  0.0004513626 -2.233664e-03 -0.0219192623 -1.719244e-02  0.005504620
## [60,] -0.2433973700 -3.242054e-02  0.0426191791 -9.412964e-03  0.011205533
## [61,] -0.0089757905 -1.877982e-01 -0.1145289030 -6.328998e-03  0.037070067
## [62,] -0.0058875871 -2.103440e-03 -0.1705212488  4.878364e-03  0.016026490
## [63,] -0.0111658616  8.596404e-03  0.0154147489 -1.457908e-01  0.045573809
## [64,] -0.0121435667 -1.073566e-03  0.0059229352 -7.913186e-03 -0.190475383
## [65,]  0.0030125445 -1.251775e-02  0.0168489770 -7.011706e-02  0.049607028
## [66,]  0.0014267339 -5.402290e-04 -0.0016107234  1.325900e-03 -0.003238096
##               [,41]         [,42]         [,43]         [,44]         [,45]
##  [1,] -0.0071563151  0.0049020592 -0.0100887688 -0.0023997818 -0.0007665168
##  [2,]  0.0068305006 -0.0037605748  0.0014168834 -0.0064290815  0.0078584040
##  [3,]  0.0026216197  0.0069187760 -0.0060747527  0.0126925609 -0.0002285053
##  [4,]  0.0063395741 -0.0018876810  0.0061902876  0.0184628318  0.0014040000
##  [5,]  0.0053641608 -0.0025442542  0.0058905711 -0.0027626719  0.0090978271
##  [6,]  0.0041508928  0.0023588900  0.0107923213  0.0029369298  0.0088585200
##  [7,]  0.0054436398 -0.0038182146  0.0031180765 -0.0138411381  0.0056422907
##  [8,]  0.0049529797  0.0048398745  0.0112549928  0.0108639596  0.0061587659
##  [9,]  0.0070004742 -0.0023487173  0.0091369241  0.0016205267  0.0026979428
## [10,] -0.0083791767  0.0045659435  0.0100106551  0.0057058968  0.0074821252
## [11,]  0.0256612638 -0.0138126159  0.0084581071 -0.0018403457  0.0009140833
## [12,] -0.0045018042  0.0225125065  0.0065538697  0.0107676836  0.0021293265
## [13,] -0.1973708965 -0.0084967589  0.0032959505 -0.0011685034  0.0045863864
## [14,] -0.0036783861  0.0229422006  0.0246423117  0.0148525847  0.0278850284
## [15,]  0.0131163041 -0.0377263264  0.0073630150 -0.0103746417 -0.0240583016
## [16,]  0.0023940657 -0.0392831733 -0.0610547475 -0.0493785320 -0.0583663073
## [17,]  0.0011179853  0.0042849727 -0.0271050894 -0.0172295521 -0.0110432953
## [18,]  0.0028571991 -0.1881755986  0.0178249914 -0.0066166265 -0.0264519577
## [19,]  0.0101627476 -0.0129189541 -0.1057689068  0.0061485308 -0.0094477197
## [20,]  0.0024071799 -0.0074191995  0.0350447889 -0.1990569694  0.0268457813
## [21,]  0.0031804235 -0.0203085896 -0.0224165809  0.0144632492 -0.1161064054
## [22,]  0.0019040978 -0.0123792830 -0.0115114011 -0.0060169846  0.0239490313
## [23,]  0.0022493200 -0.0099648974 -0.0128815509  0.0066027183 -0.0298602642
## [24,]  0.0023583189 -0.0139907556 -0.0167458254 -0.0032192523 -0.0198362859
## [25,]  0.0082140283 -0.0136775373 -0.0103490331 -0.0069999094 -0.0146233312
## [26,] -0.0116178887 -0.0040663301 -0.0023064948  0.0028604159 -0.0023132216
## [27,]  0.0281650620 -0.0145383873 -0.0025472251  0.0008563228 -0.0020198597
## [28,] -0.0588829307  0.0391820595 -0.0149110721  0.0025090579 -0.0031365303
## [29,] -0.1644545433 -0.0243602304 -0.0006268913 -0.0017684343 -0.0030199864
## [30,]  0.0360487396 -0.1611306876  0.0723154726  0.0060363454  0.0328502067
## [31,] -0.0037278812 -0.0333533923  0.0574846357  0.0397264926 -0.0002094054
## [32,] -0.0058303273  0.0175478879  0.0387729789 -0.1000291693  0.0371089727
## [33,] -0.0064714578  0.0278771256  0.0039997569  0.0602512395 -0.0541457073
## [34,] -0.0087249303  0.0255080630 -0.0088778947  0.0582120943 -0.0091270431
## [35,] -0.0060194816  0.0219887804 -0.0044334134 -0.0045916047  0.0164777512
## [36,] -0.0062772000  0.0137416280 -0.0069247166 -0.0043478509  0.0110540776
## [37,]  0.0008765050  0.0076609782  0.0013413575  0.0050784835  0.0055289481
## [38,]  0.0093037095  0.0191864843 -0.0023981827  0.0061066166  0.0139532031
## [39,]  0.0043707351  0.0037237187 -0.0104143491 -0.0026201415 -0.0001288390
## [40,]  0.0315789869  0.0351660479  0.0204149377  0.0259880075  0.0303279317
## [41,]  0.7906313811 -0.0090996537 -0.0004665397 -0.0015687401 -0.0036986728
## [42,] -0.0090996537  0.2611436653 -0.0237343739  0.0431533097  0.0376294570
## [43,] -0.0004665397 -0.0237343739  0.1273706383  0.0141832190  0.0690301272
## [44,] -0.0015687401  0.0431533097  0.0141832190  0.2504194803 -0.0298774832
## [45,] -0.0036986728  0.0376294570  0.0690301272 -0.0298774832  0.1381482709
## [46,] -0.0010236102  0.0315436258  0.0453534238  0.0659602260  0.0114090223
## [47,] -0.0009054076  0.0321238541  0.0604323202  0.0051097076  0.0844859682
## [48,] -0.0047280434  0.0286514371  0.0573956700  0.0434168008  0.0497805803
## [49,]  0.0201427622  0.0197008808  0.0398310827  0.0285668354  0.0349409605
## [50,] -0.0135540982  0.0089525113  0.0364337216  0.0270941234  0.0252371067
## [51,]  0.0172728401  0.0171769968 -0.0009707502  0.0034730694 -0.0020342586
## [52,] -0.0658959132 -0.0008440476  0.1029915521  0.0720677099  0.0841609601
## [53,] -0.1289853583  0.0179250010 -0.0290968669 -0.0143061125 -0.0206284128
## [54,] -0.0235751830  0.0907851819 -0.0040742152  0.0148474809  0.0043890186
## [55,] -0.0002979605  0.0058545660  0.0112927304  0.0113760376  0.0142110930
## [56,] -0.0061108599  0.0057620173  0.0026765830  0.0210063646  0.0370835509
## [57,] -0.0013911011 -0.0032543163  0.0277086756  0.0270820173  0.0177075765
## [58,] -0.0041415186 -0.0036339838  0.0224850654 -0.0633611767  0.0565605687
## [59,] -0.0008267653  0.0055247770  0.0319826975  0.0824263059 -0.0157441743
## [60,] -0.0151791423  0.0069055883  0.0275919752  0.0199044169  0.0203228302
## [61,]  0.0264466458  0.0151163238  0.0290769555  0.0267649221  0.0239946920
## [62,] -0.0111398923  0.0131689144  0.0363381157  0.0277184443  0.0277810204
## [63,] -0.0189216235  0.0197065409  0.0273681512  0.0212877672  0.0208865317
## [64,]  0.0438255216 -0.0088993423  0.0656012979  0.0461699033  0.0449482988
## [65,] -0.1553051621  0.0381927036  0.0311717239  0.0282117980  0.0332247909
## [66,]  0.0008584204  0.0015105475 -0.0022168498 -0.0004286292 -0.0014360714
##              [,46]         [,47]         [,48]         [,49]        [,50]
##  [1,] -0.005395759 -0.0020300767 -0.0092339567 -0.0035205649 -0.006783709
##  [2,]  0.005003038  0.0017628199  0.0052731139  0.0010156825 -0.001460847
##  [3,]  0.002905076  0.0060189298  0.0090204842  0.0032134074  0.004794522
##  [4,]  0.007664360 -0.0067145124  0.0032810350 -0.0004241638  0.002922073
##  [5,] -0.004236762  0.0142907578  0.0098012439  0.0031108232  0.002833716
##  [6,] -0.016901841  0.0080227826  0.0041290069  0.0032943612  0.002470619
##  [7,]  0.013764714  0.0143626319  0.0225004401 -0.0016330592  0.001563780
##  [8,]  0.011164500  0.0016512116 -0.0201317686 -0.0186458855 -0.001696954
##  [9,]  0.003030304  0.0049870147 -0.0100194254  0.0272998353  0.004229609
## [10,]  0.007340727  0.0029089920  0.0162104633 -0.0149281571  0.003223281
## [11,]  0.002016064  0.0022568087  0.0042752782  0.0104516890  0.006626123
## [12,]  0.008019441  0.0021294691  0.0089600506  0.0068504349 -0.003787212
## [13,]  0.001777548  0.0021385013 -0.0006985036  0.0040170280 -0.014230815
## [14,]  0.014616033  0.0180895855  0.0136578509  0.0098286963 -0.013956717
## [15,] -0.002747722 -0.0072691282  0.0073137285  0.0078062103  0.026834657
## [16,] -0.046530070 -0.0535882153 -0.0516204293 -0.0355968801 -0.028258867
## [17,] -0.017675602 -0.0200813334 -0.0258769180 -0.0211593528 -0.034025638
## [18,] -0.010844617 -0.0175145351 -0.0100341205 -0.0076975663  0.013270253
## [19,] -0.011947301  0.0055305877 -0.0024342443 -0.0004587154  0.018206844
## [20,]  0.004605268 -0.0219550007 -0.0037749758 -0.0039691978  0.013854893
## [21,] -0.024672818  0.0299041400 -0.0089357141 -0.0014176109  0.013658474
## [22,] -0.207527435  0.0418242894 -0.0080808362 -0.0128992478  0.016698192
## [23,]  0.055683160 -0.2160145831  0.0020133364  0.0387702596  0.004506044
## [24,]  0.016532852  0.0041047761 -0.1933540471  0.0272876984  0.017093321
## [25,] -0.012580567 -0.0096219537  0.0349096728 -0.1724487185  0.007365971
## [26,]  0.002289992  0.0002724940  0.0245682123 -0.0331357086 -0.202886953
## [27,] -0.002151278 -0.0010534224 -0.0160033611  0.0278007265 -0.078076279
## [28,]  0.002569589  0.0041811948 -0.0010642582  0.0208301705  0.004167632
## [29,] -0.004669085 -0.0038088680 -0.0061556046  0.0061747921  0.009241510
## [30,]  0.009476014  0.0228732917  0.0116456946  0.0054867057 -0.002614819
## [31,]  0.013474451 -0.0027339545  0.0054146579  0.0032136408 -0.005710802
## [32,]  0.014616915  0.0059869348  0.0087012249  0.0040152188 -0.001565441
## [33,]  0.015173151 -0.0306663465  0.0028419489 -0.0013017774 -0.010674306
## [34,] -0.004449759  0.0433687569 -0.0119020243 -0.0042483553 -0.014042789
## [35,]  0.022130199  0.0337406398 -0.0037179782 -0.0150233843 -0.018842030
## [36,]  0.020972838 -0.0064661729  0.0853669241  0.0251281592 -0.012488523
## [37,]  0.001659857  0.0078640833  0.0294687424  0.0582388073  0.011059602
## [38,]  0.003005054  0.0010368248 -0.0043972661 -0.0410972349  0.036506963
## [39,] -0.004218527 -0.0042237708 -0.0057877762 -0.0148714487  0.059136604
## [40,]  0.020373240  0.0265308126  0.0207839198  0.0221769574  0.035949045
## [41,] -0.001023610 -0.0009054076 -0.0047280434  0.0201427622 -0.013554098
## [42,]  0.031543626  0.0321238541  0.0286514371  0.0197008808  0.008952511
## [43,]  0.045353424  0.0604323202  0.0573956700  0.0398310827  0.036433722
## [44,]  0.065960226  0.0051097076  0.0434168008  0.0285668354  0.027094123
## [45,]  0.011409022  0.0844859682  0.0497805803  0.0349409605  0.025237107
## [46,]  0.358254624 -0.1496430285  0.0528287030  0.0225427336  0.028598854
## [47,] -0.149643029  0.3192109134  0.0257922841  0.0430258433  0.026417414
## [48,]  0.052828703  0.0257922841  0.3169376993 -0.1718341766  0.054321633
## [49,]  0.022542734  0.0430258433 -0.1718341766  0.6763679867 -0.057233221
## [50,]  0.028598854  0.0264174141  0.0543216330 -0.0572332206  0.654392923
## [51,]  0.003044453  0.0030400391 -0.0015649437  0.0159399615 -0.100481148
## [52,]  0.069205233  0.0798331217  0.0749351128  0.0595342768  0.012660339
## [53,] -0.015707972 -0.0166868093 -0.0160447958 -0.0046974743  0.016498077
## [54,]  0.012790048  0.0123520546  0.0171593610  0.0146180687  0.025321941
## [55,]  0.015965912  0.0202347404  0.0243536546  0.0198565649  0.030456931
## [56,]  0.011620482  0.0410360595  0.0187666463  0.0153273848  0.020498852
## [57,]  0.005799294  0.0228537670  0.0234969227  0.0205644760  0.032711982
## [58,]  0.137724060 -0.0414737485  0.0274631981  0.0133425920  0.026515856
## [59,] -0.067449718  0.0084577753  0.0309636715  0.0287076228  0.030702304
## [60,]  0.017520079  0.0238861747 -0.1034135125  0.0660075084  0.054924487
## [61,]  0.023609859  0.0315948896  0.0900521060 -0.1386480521 -0.069640701
## [62,]  0.025944248  0.0315353473  0.0261495255  0.0436520577 -0.243860946
## [63,]  0.019998748  0.0247256670  0.0251754821  0.0375424489  0.062864420
## [64,]  0.044093416  0.0513892009  0.0607915324  0.0250505875  0.074574283
## [65,]  0.027359132  0.0271852422  0.0371378136 -0.0280707594  0.017739497
## [66,]  0.001003144 -0.0011076230 -0.0033938698 -0.0001790917 -0.001943251
##               [,51]         [,52]         [,53]         [,54]         [,55]
##  [1,] -2.362744e-04 -0.0027672691  3.821890e-03  0.0042946818  0.0064420848
##  [2,] -3.457245e-03 -0.0029092004 -3.945623e-03 -0.0025503525 -0.0060660404
##  [3,] -2.922758e-03  0.0084405398 -5.204828e-03 -0.0050289737  0.0049222658
##  [4,]  5.607287e-06 -0.0037853405 -3.500127e-04 -0.0081621366 -0.0017895169
##  [5,] -3.654156e-03  0.0055741682 -4.663133e-03 -0.0087722711 -0.0181082767
##  [6,] -2.586034e-03  0.0045709272 -4.173618e-03 -0.0096883269 -0.0075187482
##  [7,] -5.436069e-03 -0.0009244758  4.468101e-04 -0.0088445571 -0.0122145625
##  [8,] -1.437958e-03  0.0084096630 -7.491424e-03 -0.0114954197 -0.0127565064
##  [9,] -2.854333e-03  0.0038817021 -1.297684e-03 -0.0066478285 -0.0083240307
## [10,] -2.317437e-02 -0.0031715684 -9.753400e-03 -0.0115796191 -0.0131202684
## [11,]  1.789227e-03  0.0065483779  1.664028e-02 -0.0060094521 -0.0017454146
## [12,] -1.091437e-03  0.0083584448 -2.700922e-02 -0.0103536409 -0.0263256301
## [13,]  1.484803e-02 -0.0207728917 -4.007051e-02 -0.0075554656 -0.0009902701
## [14,] -1.444509e-02  0.0137339646 -2.039883e-02 -0.0146403860 -0.0102225216
## [15,]  2.056426e-02 -0.0112653098 -2.484504e-03  0.0357974833  0.0324782154
## [16,] -6.045598e-04 -0.0837584243  2.049146e-02 -0.0102089533 -0.0179529459
## [17,] -2.581805e-02 -0.0218082657 -1.043054e-02 -0.0374298216 -0.0457777897
## [18,] -1.079595e-02  0.0484880433 -1.380166e-02 -0.0211103496  0.0253382584
## [19,]  1.714283e-02 -0.0059466407  1.616009e-02  0.0075661163 -0.0089003024
## [20,]  1.173162e-02 -0.0037041344  1.268248e-02  0.0196640860  0.0168601519
## [21,]  1.286826e-02 -0.0054391931  1.647493e-02  0.0129775794  0.0091310321
## [22,]  1.056324e-02 -0.0009103331  1.398663e-02  0.0133184210  0.0112334700
## [23,]  1.133392e-02  0.0010467682  1.228453e-02  0.0066852367  0.0053134203
## [24,]  7.291035e-03 -0.0059436468  1.538217e-02  0.0131984257  0.0091113030
## [25,]  4.123381e-02  0.0071491864  6.693775e-03  0.0152347157  0.0120282075
## [26,] -6.395587e-02 -0.0070587637  1.954917e-02  0.0069614386  0.0067163111
## [27,] -1.954139e-01 -0.0030374178  2.811197e-03 -0.0008741449 -0.0044091636
## [28,]  2.849474e-02 -0.1437866164  2.988106e-02  0.0104048649  0.0118247514
## [29,]  2.871810e-02  0.0183109324 -1.587036e-01 -0.0053938553 -0.0075883607
## [30,] -1.011305e-02  0.0297966455 -3.187831e-03 -0.1639789470 -0.0030360204
## [31,] -1.084666e-02  0.0285635859 -8.286450e-03 -0.0013752159 -0.1724986504
## [32,] -5.422098e-03  0.0179419961 -8.506838e-04 -0.0151563802  0.0391261259
## [33,] -9.145367e-03  0.0135568168  1.810981e-03 -0.0156135656 -0.0071910546
## [34,] -9.945918e-03  0.0052046923  3.659644e-03 -0.0204977955 -0.0177997091
## [35,] -1.139529e-02  0.0066780226  4.194622e-03 -0.0211957510 -0.0202140588
## [36,] -3.045840e-03 -0.0008410007  5.669321e-03 -0.0134370155 -0.0123022908
## [37,] -1.221582e-02 -0.0003687256  7.027481e-03 -0.0042795149 -0.0018785630
## [38,]  3.295186e-02  0.0205737817 -2.725252e-04 -0.0160153759 -0.0128855055
## [39,] -2.355858e-02 -0.0147251230  4.858314e-05 -0.0287178789 -0.0111604016
## [40,]  9.704179e-03 -0.0038223034 -6.233344e-02  0.0420410742  0.0063222705
## [41,]  1.727284e-02 -0.0658959132 -1.289854e-01 -0.0235751830 -0.0002979605
## [42,]  1.717700e-02 -0.0008440476  1.792500e-02  0.0907851819  0.0058545660
## [43,] -9.707502e-04  0.1029915521 -2.909687e-02 -0.0040742152  0.0112927304
## [44,]  3.473069e-03  0.0720677099 -1.430611e-02  0.0148474809  0.0113760376
## [45,] -2.034259e-03  0.0841609601 -2.062841e-02  0.0043890186  0.0142110930
## [46,]  3.044453e-03  0.0692052328 -1.570797e-02  0.0127900476  0.0159659124
## [47,]  3.040039e-03  0.0798331217 -1.668681e-02  0.0123520546  0.0202347404
## [48,] -1.564944e-03  0.0749351128 -1.604480e-02  0.0171593610  0.0243536546
## [49,]  1.593996e-02  0.0595342768 -4.697474e-03  0.0146180687  0.0198565649
## [50,] -1.004811e-01  0.0126603389  1.649808e-02  0.0253219405  0.0304569309
## [51,]  6.167288e-01  0.1315017352 -8.079108e-02  0.0220611170  0.0248276976
## [52,]  1.315017e-01  0.4806103947  2.026615e-01  0.0097027832  0.0188968745
## [53,] -8.079108e-02  0.2026615426  6.443401e-01  0.0115060952  0.0134085116
## [54,]  2.206112e-02  0.0097027832  1.150610e-02  0.0850287392  0.0272957577
## [55,]  2.482770e-02  0.0188968745  1.340851e-02  0.0272957577  0.0902033697
## [56,]  1.667382e-02  0.0182102209  1.159299e-02  0.0240298499  0.0097499458
## [57,]  2.534128e-02  0.0214996837  1.078374e-02  0.0360792252  0.0466452760
## [58,]  1.947453e-02  0.0182529171  1.096996e-02  0.0266885368  0.0303327944
## [59,]  2.429633e-02  0.0302483480  7.855727e-03  0.0366160550  0.0495093840
## [60,]  2.269597e-02  0.0270865122  4.575756e-03  0.0274958340  0.0367268752
## [61,]  7.926264e-03  0.0380427295  1.529733e-02  0.0196685261  0.0280827648
## [62,]  7.085863e-02  0.0524859794 -1.093376e-02  0.0265726714  0.0359356647
## [63,] -2.436636e-01 -0.0268345120  4.729046e-02  0.0282504564  0.0186877575
## [64,] -5.330941e-02 -0.0805106732 -9.540665e-02  0.0296481579  0.0862206352
## [65,] -8.524590e-03 -0.0896374047 -2.768191e-01  0.0216933926  0.0046344088
## [66,]  1.606402e-03 -0.0021597783 -6.278649e-04  0.0004723154 -0.0012132284
##               [,56]         [,57]         [,58]         [,59]        [,60]
##  [1,]  0.0133772438  0.0030039096  0.0086846490  0.0028012852  0.009763095
##  [2,] -0.0239183300 -0.0115227081 -0.0119222741 -0.0126097990 -0.014344408
##  [3,]  0.0015896458 -0.0084600767 -0.0061551110 -0.0084595432 -0.009704835
##  [4,] -0.0052163990 -0.0081464459 -0.0174082756 -0.0025721324 -0.012755921
##  [5,]  0.0119448206  0.0070699698  0.0017583326 -0.0231338260 -0.010177820
##  [6,] -0.0201079600 -0.0055884903 -0.0049292833  0.0118005917 -0.014893587
##  [7,]  0.0019030889 -0.0202869102  0.0185307845 -0.0076325564 -0.008274553
##  [8,] -0.0183788508 -0.0077163074 -0.0221427761 -0.0057542892  0.020352355
##  [9,] -0.0119987824 -0.0073197534 -0.0072825735 -0.0089844422  0.005172829
## [10,] -0.0140776200 -0.0113459869 -0.0147321550 -0.0089625452 -0.016295731
## [11,] -0.0095886067 -0.0009126114 -0.0029667243 -0.0014376835 -0.007346019
## [12,] -0.0219636198 -0.0213479372 -0.0264240305 -0.0193706011 -0.025968068
## [13,] -0.0020103028 -0.0028915864 -0.0002421527 -0.0038120348 -0.003332772
## [14,] -0.0114688173 -0.0116540697 -0.0136186344 -0.0007215787 -0.005946160
## [15,]  0.0018715350  0.0440417747  0.0221776199  0.0415662736  0.015578850
## [16,] -0.0149138075 -0.0168097884 -0.0139096394 -0.0249577610 -0.022324102
## [17,] -0.0287687695 -0.0495189257 -0.0372088617 -0.0501925430 -0.037991777
## [18,]  0.0165779603  0.0162943602  0.0099529411  0.0117158494  0.011260732
## [19,]  0.0220284103  0.0130501036  0.0341604678 -0.0049922760  0.010682809
## [20,] -0.0315618049  0.0207186801  0.0195172027  0.0013612345  0.006223554
## [21,]  0.0219066903 -0.0021542478 -0.0065757252  0.0093395539  0.009502452
## [22,]  0.0088412053  0.0137201121 -0.0432120886  0.0656450065  0.002585545
## [23,] -0.0028922886  0.0068514136  0.0120783811  0.0397884770  0.031462806
## [24,]  0.0090327713  0.0111463202  0.0191879069 -0.0006985245  0.037246044
## [25,]  0.0104892576  0.0145355131  0.0095008979  0.0107797611 -0.012569092
## [26,]  0.0103639422  0.0054380730  0.0107580603 -0.0014299856  0.011175159
## [27,]  0.0018058184 -0.0055586039 -0.0035121117 -0.0055745424 -0.011801786
## [28,]  0.0180973595  0.0111261284  0.0196120035 -0.0003620087  0.003673113
## [29,] -0.0032965339 -0.0092681686 -0.0077917067 -0.0073863110 -0.013043843
## [30,]  0.0136201360 -0.0169344433  0.0102696303 -0.0225637508  0.000797582
## [31,] -0.0040413081 -0.0019875815 -0.0451500487  0.0227605217 -0.001081674
## [32,] -0.2427381226  0.0204712934 -0.0180597694  0.0027094843  0.000717424
## [33,] -0.0534330167 -0.1302129716 -0.0315713937  0.0346806797 -0.004129133
## [34,] -0.0369794743  0.0344121973 -0.1937800008  0.0158360680 -0.009296833
## [35,]  0.0096030124 -0.0426987724  0.0435562481 -0.2003814410  0.016217311
## [36,]  0.0092020572 -0.0194340532  0.0110086988  0.0004513626 -0.243397370
## [37,]  0.0015175262 -0.0043879141 -0.0012329999 -0.0022336643 -0.032420540
## [38,]  0.0034578773 -0.0191297657 -0.0072106187 -0.0219192623  0.042619179
## [39,] -0.0034374389 -0.0165224158 -0.0091122265 -0.0171924383 -0.009412964
## [40,]  0.0230881477  0.0050769729  0.0143494098  0.0055046198  0.011205533
## [41,] -0.0061108599 -0.0013911011 -0.0041415186 -0.0008267653 -0.015179142
## [42,]  0.0057620173 -0.0032543163 -0.0036339838  0.0055247770  0.006905588
## [43,]  0.0026765830  0.0277086756  0.0224850654  0.0319826975  0.027591975
## [44,]  0.0210063646  0.0270820173 -0.0633611767  0.0824263059  0.019904417
## [45,]  0.0370835509  0.0177075765  0.0565605687 -0.0157441743  0.020322830
## [46,]  0.0116204818  0.0057992939  0.1377240600 -0.0674497180  0.017520079
## [47,]  0.0410360595  0.0228537670 -0.0414737485  0.0084577753  0.023886175
## [48,]  0.0187666463  0.0234969227  0.0274631981  0.0309636715 -0.103413512
## [49,]  0.0153273848  0.0205644760  0.0133425920  0.0287076228  0.066007508
## [50,]  0.0204988517  0.0327119817  0.0265158560  0.0307023039  0.054924487
## [51,]  0.0166738205  0.0253412836  0.0194745332  0.0242963297  0.022695973
## [52,]  0.0182102209  0.0214996837  0.0182529171  0.0302483480  0.027086512
## [53,]  0.0115929903  0.0107837419  0.0109699594  0.0078557273  0.004575756
## [54,]  0.0240298499  0.0360792252  0.0266885368  0.0366160550  0.027495834
## [55,]  0.0097499458  0.0466452760  0.0303327944  0.0495093840  0.036726875
## [56,]  0.1766257243  0.0016891977  0.1042088663 -0.0311660184  0.028886631
## [57,]  0.0016891977  0.0930744302 -0.0089448933  0.0746616531  0.036864391
## [58,]  0.1042088663 -0.0089448933  0.2283837752 -0.0904729594  0.031572093
## [59,] -0.0311660184  0.0746616531 -0.0904729594  0.2204498799  0.021043413
## [60,]  0.0288866311  0.0368643907  0.0315720935  0.0210434133  0.216196443
## [61,]  0.0210039519  0.0277969788  0.0225737814  0.0378445120 -0.084958667
## [62,]  0.0257939524  0.0369385209  0.0289823520  0.0412905709  0.028776885
## [63,]  0.0128229667  0.0213392381  0.0159289278  0.0252632023  0.003021535
## [64,]  0.0595270804  0.0845595315  0.0675439879  0.0899122647  0.085553199
## [65,]  0.0083739845  0.0054863219  0.0068501128  0.0066022937  0.037034583
## [66,] -0.0007118702 -0.0006382275 -0.0007119330 -0.0014412263 -0.001437497
##               [,61]         [,62]         [,63]         [,64]         [,65]
##  [1,]  0.0104044142  5.728068e-03 -3.419157e-03  0.0209608083  0.0003527435
##  [2,] -0.0115929036 -9.374377e-03 -8.943102e-03 -0.0085647920 -0.0113280199
##  [3,] -0.0094977385 -6.571403e-03  6.241569e-03 -0.0323573398  0.0067236546
##  [4,] -0.0118405742 -9.370131e-03 -1.684691e-03 -0.0239848394 -0.0026816631
##  [5,] -0.0101525183 -7.886450e-03  1.929349e-03 -0.0266088419  0.0012888016
##  [6,] -0.0110782960 -7.835932e-03  8.814070e-05 -0.0233372402  0.0024200813
##  [7,] -0.0128007927 -9.636796e-03 -3.332741e-04 -0.0283055253 -0.0031413913
##  [8,]  0.0151152760 -8.658827e-03  2.141076e-03 -0.0263821304  0.0025387744
##  [9,] -0.0149840042 -1.366513e-03 -4.951968e-03 -0.0254267385 -0.0009411105
## [10,] -0.0028193023 -7.409708e-03  1.829574e-02 -0.0111549692  0.0163543524
## [11,] -0.0016943779  5.255333e-03  1.595621e-02 -0.0209826817 -0.0122243577
## [12,] -0.0145975989 -1.630255e-02 -1.128470e-02 -0.0302887758  0.0138754128
## [13,]  0.0064127982 -1.735602e-03 -1.017475e-02  0.0099381145  0.0517188715
## [14,]  0.0043901875  7.247816e-03  5.292399e-03  0.0045922966  0.0104690932
## [15,]  0.0003724539  1.765083e-02  1.901350e-02  0.0347588717 -0.0104445371
## [16,] -0.0269419998 -3.115579e-02 -2.384871e-02 -0.0514272595 -0.0325452967
## [17,] -0.0275585569 -3.766240e-02 -2.247217e-02 -0.0852791175 -0.0057663702
## [18,]  0.0015576461 -1.210974e-03 -8.640609e-03  0.0248858203 -0.0169420809
## [19,]  0.0006178148  1.705025e-03 -2.399805e-03  0.0185187425 -0.0104712607
## [20,] -0.0005271324 -1.098122e-03 -4.518773e-06  0.0022499661 -0.0038651481
## [21,] -0.0021158524 -3.345845e-03 -2.154355e-03  0.0007927347 -0.0076411454
## [22,]  0.0014793587 -2.520089e-03 -1.026163e-03  0.0036390938 -0.0031114287
## [23,] -0.0198264225 -3.438155e-03 -4.365230e-03 -0.0058656088 -0.0050387393
## [24,] -0.0089147907 -5.729733e-03 -1.601205e-03  0.0006104786 -0.0043066024
## [25,]  0.0305692711  1.200857e-02 -1.394701e-02 -0.0033836824 -0.0116242270
## [26,] -0.0130678251  3.182974e-02  5.828399e-02  0.0384247868  0.0251602559
## [27,]  0.0293778669  5.752774e-02  5.152550e-02 -0.0030166177 -0.0111205966
## [28,]  0.0284116775  1.574094e-02  3.741232e-03  0.0263098612 -0.0001738525
## [29,]  0.0161002331 -2.172031e-02 -2.030149e-02 -0.0169109756  0.0243445610
## [30,]  0.0063786523  1.250982e-04 -1.627266e-02  0.0355385629 -0.0082050842
## [31,]  0.0074810063  1.265884e-03 -6.750568e-03  0.0114439363  0.0023271543
## [32,]  0.0057913246  1.427160e-03 -6.721685e-04  0.0035841644  0.0084661544
## [33,]  0.0074808361 -3.785482e-03 -6.847785e-03 -0.0067480539  0.0069955684
## [34,] -0.0041563878 -1.050873e-02 -1.293896e-02 -0.0181538508  0.0074793219
## [35,]  0.0176569382 -1.027706e-02 -9.413285e-03 -0.0229148853  0.0018986146
## [36,] -0.0089757905 -5.887587e-03 -1.116586e-02 -0.0121435667  0.0030125445
## [37,] -0.1877981617 -2.103440e-03  8.596404e-03 -0.0010735658 -0.0125177505
## [38,] -0.1145289030 -1.705212e-01  1.541475e-02  0.0059229352  0.0168489770
## [39,] -0.0063289980  4.878364e-03 -1.457908e-01 -0.0079131859 -0.0701170645
## [40,]  0.0370700674  1.602649e-02  4.557381e-02 -0.1904753829  0.0496070281
## [41,]  0.0264466458 -1.113989e-02 -1.892162e-02  0.0438255216 -0.1553051621
## [42,]  0.0151163238  1.316891e-02  1.970654e-02 -0.0088993423  0.0381927036
## [43,]  0.0290769555  3.633812e-02  2.736815e-02  0.0656012979  0.0311717239
## [44,]  0.0267649221  2.771844e-02  2.128777e-02  0.0461699033  0.0282117980
## [45,]  0.0239946920  2.778102e-02  2.088653e-02  0.0449482988  0.0332247909
## [46,]  0.0236098586  2.594425e-02  1.999875e-02  0.0440934156  0.0273591321
## [47,]  0.0315948896  3.153535e-02  2.472567e-02  0.0513892009  0.0271852422
## [48,]  0.0900521060  2.614953e-02  2.517548e-02  0.0607915324  0.0371378136
## [49,] -0.1386480521  4.365206e-02  3.754245e-02  0.0250505875 -0.0280707594
## [50,] -0.0696407009 -2.438609e-01  6.286442e-02  0.0745742832  0.0177394965
## [51,]  0.0079262635  7.085863e-02 -2.436636e-01 -0.0533094141 -0.0085245905
## [52,]  0.0380427295  5.248598e-02 -2.683451e-02 -0.0805106732 -0.0896374047
## [53,]  0.0152973263 -1.093376e-02  4.729046e-02 -0.0954066549 -0.2768190722
## [54,]  0.0196685261  2.657267e-02  2.825046e-02  0.0296481579  0.0216933926
## [55,]  0.0280827648  3.593566e-02  1.868776e-02  0.0862206352  0.0046344088
## [56,]  0.0210039519  2.579395e-02  1.282297e-02  0.0595270804  0.0083739845
## [57,]  0.0277969788  3.693852e-02  2.133924e-02  0.0845595315  0.0054863219
## [58,]  0.0225737814  2.898235e-02  1.592893e-02  0.0675439879  0.0068501128
## [59,]  0.0378445120  4.129057e-02  2.526320e-02  0.0899122647  0.0066022937
## [60,] -0.0849586665  2.877689e-02  3.021535e-03  0.0855531992  0.0370345834
## [61,]  0.4418746921  4.445450e-02  5.836922e-02  0.0357137049 -0.0649127298
## [62,]  0.0444545002  2.133375e-01 -2.879642e-02  0.0718278618  0.0397608647
## [63,]  0.0583692175 -2.879642e-02  1.794697e-01  0.0511025888  0.0086370152
## [64,]  0.0357137049  7.182786e-02  5.110259e-02  0.3559934776  0.0913892586
## [65,] -0.0649127298  3.976086e-02  8.637015e-03  0.0913892586  0.2765607019
## [66,]  0.0001890885  5.190561e-05 -1.735812e-03 -0.0020379262 -0.0008997256
##               [,66]
##  [1,]  1.158441e-03
##  [2,] -1.973227e-04
##  [3,] -1.527481e-03
##  [4,] -4.290852e-04
##  [5,] -1.951885e-03
##  [6,] -1.736935e-03
##  [7,] -1.938593e-03
##  [8,] -5.927517e-04
##  [9,] -1.644957e-03
## [10,] -3.278450e-04
## [11,] -2.822354e-03
## [12,] -1.529455e-03
## [13,]  1.338728e-03
## [14,]  9.941847e-04
## [15,]  6.513935e-04
## [16,]  1.170427e-03
## [17,]  7.823946e-04
## [18,] -3.350941e-03
## [19,]  4.903678e-04
## [20,] -1.845255e-04
## [21,] -2.066360e-03
## [22,] -2.290041e-03
## [23,] -3.596957e-03
## [24,]  1.722101e-03
## [25,]  2.471111e-03
## [26,] -6.296649e-03
## [27,]  2.228085e-03
## [28,] -6.468791e-04
## [29,] -2.058077e-03
## [30,] -4.136589e-03
## [31,]  1.297019e-03
## [32,] -2.637253e-03
## [33,] -8.377401e-05
## [34,]  2.258945e-04
## [35,]  7.066990e-04
## [36,]  1.426734e-03
## [37,] -5.402290e-04
## [38,] -1.610723e-03
## [39,]  1.325900e-03
## [40,] -3.238096e-03
## [41,]  8.584204e-04
## [42,]  1.510547e-03
## [43,] -2.216850e-03
## [44,] -4.286292e-04
## [45,] -1.436071e-03
## [46,]  1.003144e-03
## [47,] -1.107623e-03
## [48,] -3.393870e-03
## [49,] -1.790917e-04
## [50,] -1.943251e-03
## [51,]  1.606402e-03
## [52,] -2.159778e-03
## [53,] -6.278649e-04
## [54,]  4.723154e-04
## [55,] -1.213228e-03
## [56,] -7.118702e-04
## [57,] -6.382275e-04
## [58,] -7.119330e-04
## [59,] -1.441226e-03
## [60,] -1.437497e-03
## [61,]  1.890885e-04
## [62,]  5.190561e-05
## [63,] -1.735812e-03
## [64,] -2.037926e-03
## [65,] -8.997256e-04
## [66,]  5.342066e-03
## 
## $log_evidence
## [1] -160.9416
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 136

3g)

Use the viz_post_coefs() function to visualize the posterior coefficient summaries for model 3 and model 6, based on the strong prior specification.

SOLUTION

post_means_3_1 <- laplace_03_strong$mode[1:ncol(X03)]
post_sds_3_1 <- sqrt(diag(laplace_03_strong$var_matrix)[1:ncol(X03)])
viz_post_coefs(post_means_3_1, post_sds_3_1, xnames_3)

post_means_6_1 <- laplace_06_strong$mode[1:ncol(X06)]
post_sds_6_1 <- sqrt(diag(laplace_06_strong$var_matrix)[1:ncol(X06)])
viz_post_coefs(post_means_6_1, post_sds_6_1, xnames_6)

### 3h)

You will fit one more set of Bayesian models with a very strong prior on the regression coefficients. The prior standard deviation will be equal to 1/50.

Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the very strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.

Run the Laplace Approximation using the strong prior for both the model 3 and model 6 formulations. Assign the results to laplace_03_very_strong and laplace_06_very_strong.

Confirm that the optimizations converged for both laplace approximation results.

SOLUTION

info_03_very_strong <- list(
  yobs = df$y ,
  design_matrix = X03,
  mu_beta = 0,
  tau_beta = 1/50,
  sigma_rate = 1
)

info_06_very_strong <- list(
  yobs = df$y,
  design_matrix = X06,
  mu_beta = 0,
  tau_beta = 1/50,
  sigma_rate = 1
)

Execute the Laplace Approximation.

laplace_03_very_strong <- my_laplace(rep(0, ncol(X03)+1), lm_logpost, info_03_very_strong)
laplace_03_very_strong

## $mode
##  [1]  0.003592816  0.006873373 -0.003563880 -0.005548824 -0.034591622
##  [6] -0.005597032  0.006696299 -0.010922109 -0.036247874 -0.082665512
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  3.845484e-04  2.590441e-07 -1.379062e-05 -2.342354e-07 -1.718063e-05
##  [2,]  2.590441e-07  3.840058e-04  3.516810e-06 -1.839586e-06 -3.016304e-06
##  [3,] -1.379062e-05  3.516810e-06  3.644930e-04 -1.862476e-06 -1.159400e-05
##  [4,] -2.342354e-07 -1.839586e-06 -1.862476e-06  3.837901e-04  2.426885e-06
##  [5,] -1.718063e-05 -3.016304e-06 -1.159400e-05  2.426885e-06  3.747607e-04
##  [6,] -9.397066e-07 -3.102059e-06 -3.791049e-06  2.222883e-06 -5.962894e-06
##  [7,]  1.343445e-06 -1.574856e-05  6.109597e-06 -9.945576e-06  2.864430e-06
##  [8,] -3.066294e-06 -6.377982e-06 -2.351642e-06 -1.551118e-05  6.543867e-07
##  [9,] -1.471934e-05  2.996232e-06 -3.702898e-05 -2.145648e-06 -2.186025e-05
## [10,] -6.498831e-05 -6.513476e-05 -1.287704e-05  5.111107e-05  3.033540e-04
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,] -9.397066e-07  1.343445e-06 -3.066294e-06 -1.471934e-05 -6.498831e-05
##  [2,] -3.102059e-06 -1.574856e-05 -6.377982e-06  2.996232e-06 -6.513476e-05
##  [3,] -3.791049e-06  6.109597e-06 -2.351642e-06 -3.702898e-05 -1.287704e-05
##  [4,]  2.222883e-06 -9.945576e-06 -1.551118e-05 -2.145648e-06  5.111107e-05
##  [5,] -5.962894e-06  2.864430e-06  6.543867e-07 -2.186025e-05  3.033540e-04
##  [6,]  3.826461e-04 -6.989646e-06  9.383095e-06 -1.666291e-05  3.746312e-05
##  [7,] -6.989646e-06  3.529481e-04 -2.161345e-05  8.768167e-06 -5.358861e-05
##  [8,]  9.383095e-06 -2.161345e-05  3.545232e-04  2.842754e-06  1.034925e-04
##  [9,] -1.666291e-05  8.768167e-06  2.842754e-06  3.207897e-04  2.649719e-04
## [10,]  3.746312e-05 -5.358861e-05  1.034925e-04  2.649719e-04  5.546765e-03
## 
## $log_evidence
## [1] -140.2491
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 76

laplace_06_very_strong <- my_laplace(rep(0, ncol(X06)+1), lm_logpost, info_06_very_strong)
laplace_06_very_strong

## $mode
##  [1]  9.596943e-03  2.025636e-04  4.495974e-04  1.246452e-03  2.046718e-03
##  [6]  1.146167e-03  2.338609e-04  1.993863e-03  1.798106e-03  9.420842e-04
## [11]  1.640120e-03 -8.309383e-05  2.354423e-04 -7.060179e-03 -1.630535e-02
## [16] -1.278185e-02 -5.099149e-02 -8.855226e-04  9.591070e-04 -1.883520e-04
## [21] -4.065259e-04  2.894725e-04 -2.912267e-04 -1.957848e-03 -1.221818e-03
## [26] -1.863038e-04  5.372941e-05 -1.753379e-03  3.442692e-04 -8.925697e-05
## [31] -4.985172e-04 -1.637362e-04 -3.140736e-03 -2.410259e-03 -1.819216e-03
## [36] -7.196701e-04  3.763246e-05 -5.751050e-04 -2.466661e-04 -2.427530e-03
## [41]  3.903263e-04 -3.534247e-03  3.819679e-03 -3.536720e-04  8.327978e-04
## [46]  1.454220e-03 -1.924209e-03 -2.628542e-03 -1.208920e-03 -1.183356e-03
## [51] -7.888923e-04 -4.438683e-03  7.272193e-04  5.321576e-03  6.989395e-04
## [56] -9.418775e-04 -1.477772e-02 -1.059962e-02 -5.658545e-03 -3.733996e-03
## [61] -1.390625e-03 -2.460287e-03 -2.553944e-03 -6.012999e-03  1.946399e-04
## [66] -1.902831e-01
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  3.837139e-04 -1.519321e-06 -1.564882e-06 -1.076749e-06 -1.257282e-06
##  [2,] -1.519321e-06  3.991731e-04 -4.445632e-07 -1.235545e-08  5.999702e-08
##  [3,] -1.564882e-06 -4.445632e-07  3.989231e-04 -3.276401e-07  2.571449e-08
##  [4,] -1.076749e-06 -1.235545e-08 -3.276401e-07  3.993237e-04 -3.817928e-07
##  [5,] -1.257282e-06  5.999702e-08  2.571449e-08 -3.817928e-07  3.989649e-04
##  [6,] -1.109776e-06  4.152320e-08  4.574653e-08  4.069365e-08 -3.592712e-07
##  [7,] -1.250205e-06  2.339550e-08  2.508779e-08  2.675182e-08  3.756982e-09
##  [8,] -1.834348e-06  3.848942e-08  4.994831e-08  7.862574e-08  1.358519e-07
##  [9,] -1.249469e-06  2.654528e-08  3.666408e-08  6.377212e-08  1.107183e-07
## [10,] -1.207298e-06  3.377763e-08  4.033826e-08  5.276396e-08  8.857227e-08
## [11,] -2.984418e-07  1.809489e-07  4.784079e-08  6.885909e-08  1.173619e-07
## [12,] -1.676746e-06 -3.308902e-07  1.674387e-08  2.450361e-08  3.775493e-08
## [13,]  3.838930e-07  2.073298e-07  5.204072e-09  2.884237e-09  5.113285e-09
## [14,] -8.550244e-07 -6.832992e-08  2.559590e-07 -3.131557e-07 -8.770412e-08
## [15,] -1.289708e-05 -7.655103e-07 -9.778822e-07 -9.142998e-07 -1.632681e-06
## [16,] -5.381599e-06  3.676965e-07 -2.667658e-07 -4.497462e-07  3.740820e-07
## [17,] -1.674575e-05 -1.778038e-06 -1.544202e-06 -1.242393e-06 -2.595720e-06
## [18,] -5.750375e-07 -6.388654e-08  2.977858e-08 -4.156419e-08 -1.103108e-07
## [19,]  6.691208e-07  1.101250e-07  2.470961e-07  3.684964e-08  3.930172e-08
## [20,] -1.667181e-07  8.788874e-09  4.976082e-10 -1.320205e-07 -7.651909e-08
## [21,] -1.576359e-07 -4.657183e-08 -4.760074e-08 -1.065255e-07  6.655278e-08
## [22,]  2.135272e-07 -2.217752e-08 -1.357117e-08 -7.481108e-09  1.454594e-07
## [23,] -1.467427e-07  6.153365e-09  7.647601e-09  5.037906e-09  1.330210e-08
## [24,]  1.796292e-07 -8.322053e-09 -1.707008e-08 -4.730141e-08 -8.929771e-08
## [25,]  2.113376e-07 -6.420554e-09 -1.238423e-08 -3.168123e-08 -5.813182e-08
## [26,]  2.305971e-07  3.659493e-08  3.794353e-08  3.333505e-08  5.261776e-08
## [27,]  1.416995e-07  7.252342e-08  4.696737e-08  4.327278e-08  7.285785e-08
## [28,] -8.428031e-07 -6.210066e-08 -1.041957e-09 -2.613757e-08 -5.768223e-08
## [29,]  6.629177e-08  4.842842e-08  1.018248e-08  1.242769e-08  2.509570e-08
## [30,] -2.068694e-08 -6.364691e-07 -1.689326e-07  7.498691e-08  2.557167e-07
## [31,] -6.934362e-07 -2.509855e-07 -9.715332e-07 -2.225146e-07  1.284143e-07
## [32,] -6.556152e-07  1.371652e-08 -2.455642e-07 -4.316792e-07 -3.438460e-07
## [33,] -1.680073e-06  1.197399e-07  7.260871e-08 -4.056919e-07 -1.265041e-06
## [34,] -1.648618e-06  7.303616e-08  5.210023e-08 -1.049083e-08 -4.872420e-07
## [35,] -1.489113e-06  2.992818e-08  2.126823e-08 -1.251886e-08 -5.861654e-08
## [36,] -1.485716e-06  2.933567e-08  2.715551e-08  1.060371e-08  1.463767e-08
## [37,] -6.111615e-07  1.764508e-08  1.868058e-08  1.694841e-08  2.824581e-08
## [38,] -4.960276e-07  7.918334e-08  8.092667e-08  6.500020e-08  1.036678e-07
## [39,] -6.098736e-08  1.847208e-07  7.662625e-08  6.637148e-08  1.045900e-07
## [40,] -1.422680e-06 -2.115330e-07  4.941677e-08  5.765748e-09  4.617188e-09
## [41,]  2.122005e-07  1.470532e-07 -9.390409e-09 -3.158401e-09 -8.084341e-09
## [42,] -3.385838e-06  2.224558e-07 -3.590062e-07 -3.513994e-07 -7.247000e-07
## [43,]  1.596640e-06  2.573600e-07  6.565460e-08  8.925257e-08  1.813037e-07
## [44,] -6.452458e-08 -3.003364e-09 -5.889629e-08 -1.107412e-07 -6.305295e-09
## [45,] -3.836549e-08 -1.882791e-07 -1.554138e-07 -9.362926e-08  4.525258e-07
## [46,]  5.649393e-07 -8.476549e-08 -5.263209e-08 -3.404431e-08  5.003071e-07
## [47,] -5.661797e-07  1.431463e-08  1.170743e-08 -1.762456e-08 -1.592168e-08
## [48,] -4.665317e-07  1.326987e-08  3.609210e-09 -3.951104e-08 -8.094951e-08
## [49,] -9.887992e-08  1.077033e-08  6.296524e-09 -1.425080e-08 -3.045701e-08
## [50,] -7.369209e-08  1.558924e-07  1.592102e-07  1.286692e-07  2.047467e-07
## [51,]  2.071709e-07  2.228181e-07  1.949602e-07  1.626013e-07  2.738781e-07
## [52,] -2.038103e-06 -7.384519e-08 -9.526350e-09 -7.110338e-08 -1.640892e-07
## [53,]  3.406853e-07  8.504126e-08  4.122186e-08  4.218652e-08  8.736454e-08
## [54,]  6.654122e-06 -2.006310e-06  3.074057e-07  8.614761e-07  2.060453e-06
## [55,]  5.877754e-07 -6.771396e-07 -2.749600e-06 -3.449398e-07  8.684285e-07
## [56,] -1.225256e-06  7.376936e-08 -6.448106e-07 -9.709896e-07 -8.948254e-07
## [57,] -5.018402e-06  4.355426e-07  2.710203e-07 -1.173946e-06 -3.798048e-06
## [58,] -4.221252e-06  2.368078e-07  1.471119e-07 -8.083826e-08 -1.500448e-06
## [59,] -2.988899e-06  7.226641e-08  4.569579e-08 -5.569023e-08 -1.748225e-07
## [60,] -2.408519e-06  5.283806e-08  3.813737e-08 -3.149796e-08 -6.805998e-08
## [61,] -9.087086e-07  3.642180e-08  3.189334e-08  4.311014e-09  1.152071e-09
## [62,] -4.699933e-07  3.232340e-07  3.296436e-07  2.646690e-07  4.230983e-07
## [63,] -3.936609e-07  5.705443e-07  2.722229e-07  2.192270e-07  3.274238e-07
## [64,] -1.870912e-06 -4.238568e-07  2.946487e-07  1.478002e-07  2.671780e-07
## [65,] -2.623988e-07  3.282216e-07 -1.039126e-07 -8.065800e-08 -1.634222e-07
## [66,] -1.260574e-04 -3.626361e-06 -6.311082e-06 -1.493025e-05 -2.623247e-05
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,] -1.109776e-06 -1.250205e-06 -1.834348e-06 -1.249469e-06 -1.207298e-06
##  [2,]  4.152320e-08  2.339550e-08  3.848942e-08  2.654528e-08  3.377763e-08
##  [3,]  4.574653e-08  2.508779e-08  4.994831e-08  3.666408e-08  4.033826e-08
##  [4,]  4.069365e-08  2.675182e-08  7.862574e-08  6.377212e-08  5.276396e-08
##  [5,] -3.592712e-07  3.756982e-09  1.358519e-07  1.107183e-07  8.857227e-08
##  [6,]  3.992701e-04 -3.652864e-07  2.221968e-08  6.800429e-08  5.642474e-08
##  [7,] -3.652864e-07  3.993190e-04 -3.690865e-07 -1.907561e-09  2.952867e-08
##  [8,]  2.221968e-08 -3.690865e-07  3.986083e-04 -5.015527e-07  3.593506e-08
##  [9,]  6.800429e-08 -1.907561e-09 -5.015527e-07  3.992333e-04 -2.951211e-07
## [10,]  5.642474e-08  2.952867e-08  3.593506e-08 -2.951211e-07  3.992292e-04
## [11,]  7.153409e-08  3.075759e-08  9.561291e-08  5.273807e-08 -3.729151e-07
## [12,]  2.740392e-08  2.265035e-08  2.862501e-08  9.975837e-09 -1.258796e-07
## [13,]  1.895879e-09 -1.452365e-09  5.529101e-09  1.087090e-08  4.180457e-08
## [14,]  9.153257e-08 -2.122896e-07  3.381300e-07  2.605445e-07 -1.477257e-07
## [15,] -1.413540e-06 -1.072335e-06 -1.547007e-06 -8.666124e-07 -8.540470e-07
## [16,]  2.950458e-07 -5.040960e-07 -6.186057e-07 -3.995322e-07 -1.520142e-06
## [17,] -1.698675e-06 -8.595517e-07 -1.176875e-06 -7.766387e-07 -1.597907e-06
## [18,] -7.183237e-08 -1.870980e-08 -5.463448e-08 -4.328024e-08 -3.028541e-08
## [19,]  2.480451e-08  1.741412e-08  5.040355e-08  4.201761e-08  5.072285e-08
## [20,] -8.415158e-09  4.808187e-09 -3.180114e-09 -4.449448e-09  1.088968e-08
## [21,]  1.106016e-07 -9.773000e-09 -4.770929e-08 -3.686993e-08 -1.901321e-08
## [22,]  1.734109e-07 -4.358655e-08 -2.322016e-08 -2.404004e-10  6.596301e-09
## [23,] -3.273093e-08 -1.737268e-07 -4.131356e-08  5.624809e-09  1.587341e-08
## [24,] -7.154553e-08 -5.151375e-08  3.156510e-07  1.411342e-07 -2.242782e-08
## [25,] -3.400915e-08  7.635810e-11  1.599008e-07  1.585759e-07  2.066964e-08
## [26,]  3.557977e-08  2.886154e-08  4.143373e-08  6.363559e-08 -5.299725e-08
## [27,]  4.845908e-08  3.255468e-08  4.749011e-08  3.421820e-08 -6.824987e-08
## [28,] -3.076281e-08  7.841754e-09 -4.677624e-08 -4.647118e-08 -4.843687e-08
## [29,]  1.558277e-08  3.937542e-09  1.675159e-08  1.419238e-08  1.731877e-08
## [30,]  1.755555e-07  8.259817e-08  1.043146e-07  6.302210e-08  1.144692e-07
## [31,]  9.985607e-08  6.482915e-08  7.050605e-08  3.883455e-08  9.726996e-08
## [32,]  4.576997e-08  3.702257e-08  4.912109e-08  3.105309e-08  5.466462e-08
## [33,] -4.400859e-07  2.057449e-08 -8.131006e-09 -3.027241e-08  3.106650e-08
## [34,] -8.387073e-07 -4.895062e-07 -8.284154e-08 -3.213951e-08  1.348251e-08
## [35,] -5.307712e-07 -7.173043e-07 -2.813866e-07 -5.765654e-08  6.769729e-09
## [36,] -4.936159e-08 -2.293525e-07 -1.044730e-06 -3.582164e-07  1.730598e-08
## [37,]  2.035787e-08 -6.138817e-09 -3.356500e-07 -2.737345e-07 -1.291919e-07
## [38,]  7.239620e-08  6.055110e-08  5.929561e-08 -1.117459e-07 -6.463108e-07
## [39,]  7.057854e-08  5.704087e-08  7.272529e-08  2.984311e-08 -3.100082e-07
## [40,]  1.612564e-08  3.951713e-08 -2.460434e-08 -4.478629e-08 -1.050419e-07
## [41,] -7.937527e-09 -6.724786e-09  3.538177e-09  1.028622e-08  4.032289e-08
## [42,] -4.829355e-07 -1.604540e-07 -3.245248e-07 -2.370370e-07 -2.383910e-07
## [43,]  9.467207e-08  7.353569e-08  2.116831e-07  1.751692e-07  2.094606e-07
## [44,] -3.096916e-08  1.471957e-08 -3.656012e-09 -8.063586e-09  3.966682e-08
## [45,]  3.854712e-07 -6.172124e-08 -1.075114e-07 -6.924746e-08 -5.031207e-08
## [46,]  4.610482e-07 -1.600412e-07 -4.380679e-08  7.612283e-09  1.454454e-08
## [47,] -1.400024e-07 -3.627030e-07 -1.513211e-07 -5.047498e-08  1.705049e-08
## [48,] -8.002421e-08 -9.990984e-08 -2.459550e-07 -5.083184e-08 -1.867701e-09
## [49,] -1.440463e-08  2.731475e-09 -6.941742e-09  1.895353e-08 -1.179737e-07
## [50,]  1.419161e-07  1.190931e-07  1.341218e-07 -4.287190e-08 -8.236324e-07
## [51,]  1.871972e-07  1.352034e-07  1.640622e-07  8.424377e-08 -3.738133e-07
## [52,] -9.120650e-08  2.135625e-08 -1.126768e-07 -1.189516e-07 -2.045166e-07
## [53,]  5.657706e-08  1.691866e-08  4.593994e-08  4.080069e-08  9.051308e-08
## [54,]  1.382405e-06  5.707265e-07  9.098340e-07  6.159386e-07  8.508736e-07
## [55,]  5.905910e-07  3.371202e-07  4.635837e-07  2.966482e-07  5.511678e-07
## [56,]  1.715754e-07  1.270496e-07  1.555114e-07  9.432672e-08  1.873502e-07
## [57,] -1.273372e-06  1.189811e-07 -1.687243e-07 -2.337995e-07  2.206129e-08
## [58,] -1.992729e-06 -1.002391e-06 -2.957081e-07 -2.035277e-07 -1.915303e-08
## [59,] -1.150191e-06 -1.396940e-06 -4.667180e-07 -1.623801e-07  3.300283e-09
## [60,] -1.513708e-07 -3.140392e-07 -1.624907e-06 -5.619991e-07  1.722302e-08
## [61,]  1.002863e-08 -4.275814e-09 -4.882826e-07 -3.025212e-07 -2.901002e-07
## [62,]  2.939104e-07  2.455000e-07  2.660183e-07 -1.575100e-07 -1.871614e-06
## [63,]  2.271826e-07  2.191335e-07  2.300484e-07  9.387797e-08 -9.293262e-07
## [64,]  2.132891e-07  1.846603e-07  5.536494e-08 -3.538134e-08 -2.251565e-07
## [65,] -1.137247e-07 -5.225543e-08 -6.269617e-08 -2.644769e-08  7.304006e-08
## [66,] -1.526203e-05 -4.300049e-06 -2.218876e-05 -1.939114e-05 -1.241618e-05
##               [,11]         [,12]         [,13]         [,14]         [,15]
##  [1,] -2.984418e-07 -1.676746e-06  3.838930e-07 -8.550244e-07 -1.289708e-05
##  [2,]  1.809489e-07 -3.308902e-07  2.073298e-07 -6.832992e-08 -7.655103e-07
##  [3,]  4.784079e-08  1.674387e-08  5.204072e-09  2.559590e-07 -9.778822e-07
##  [4,]  6.885909e-08  2.450361e-08  2.884237e-09 -3.131557e-07 -9.142998e-07
##  [5,]  1.173619e-07  3.775493e-08  5.113285e-09 -8.770412e-08 -1.632681e-06
##  [6,]  7.153409e-08  2.740392e-08  1.895879e-09  9.153257e-08 -1.413540e-06
##  [7,]  3.075759e-08  2.265035e-08 -1.452365e-09 -2.122896e-07 -1.072335e-06
##  [8,]  9.561291e-08  2.862501e-08  5.529101e-09  3.381300e-07 -1.547007e-06
##  [9,]  5.273807e-08  9.975837e-09  1.087090e-08  2.605445e-07 -8.666124e-07
## [10,] -3.729151e-07 -1.258796e-07  4.180457e-08 -1.477257e-07 -8.540470e-07
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## [13,] -1.333018e-07  1.350747e-07  3.996268e-04 -6.693949e-08  1.364972e-07
## [14,] -4.034315e-07 -5.447775e-07 -6.693949e-08  3.854277e-04 -5.797234e-07
## [15,] -6.337537e-07 -1.025171e-06  1.364972e-07 -5.797234e-07  3.801979e-04
## [16,] -1.686009e-06 -1.190856e-06 -6.151977e-08 -3.379213e-05 -6.456675e-06
## [17,] -1.716648e-06 -1.243791e-06  1.264394e-07 -8.805917e-07 -5.313877e-05
## [18,] -7.172254e-09 -8.914043e-08  4.791025e-08 -1.161310e-06 -3.585316e-07
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## [20,]  9.369027e-09  1.452754e-08 -7.452063e-10 -9.849688e-07  5.645669e-10
## [21,] -2.483240e-08 -3.007519e-09  1.541119e-09 -1.760698e-06  6.145698e-07
## [22,]  1.242561e-08  3.620781e-09  3.570675e-09 -1.607377e-06  4.636755e-07
## [23,]  1.404120e-08  1.926995e-08 -1.462228e-09 -1.256562e-06 -2.052442e-07
## [24,] -5.485238e-08  9.563913e-09 -8.460010e-09 -1.253758e-06  3.236918e-07
## [25,] -3.564940e-08  3.818931e-09 -5.153329e-09 -5.729760e-07  2.776068e-07
## [26,] -7.381944e-08 -1.410182e-09  6.167946e-09 -9.838415e-07 -6.433824e-07
## [27,] -2.595183e-07 -9.924655e-09 -5.159390e-08 -6.017925e-07 -6.803010e-07
## [28,] -8.238001e-08 -2.452305e-07 -8.279749e-09 -1.196349e-06 -7.070486e-07
## [29,] -3.672564e-08  1.560409e-09 -2.167409e-07  2.077234e-07 -5.367791e-08
## [30,]  2.172762e-07 -1.628832e-07  1.373135e-07  1.847011e-07 -8.150956e-07
## [31,]  9.170936e-08  8.584167e-08 -8.496671e-09  3.319922e-07 -2.023334e-06
## [32,]  5.261459e-08  5.004448e-08 -6.533962e-09 -1.234361e-07 -1.640479e-06
## [33,] -1.077104e-08  7.234099e-08 -2.455746e-08  1.134604e-06 -2.840721e-06
## [34,] -2.093552e-08  4.963683e-08 -1.812585e-08  9.067545e-07 -2.254085e-06
## [35,] -2.034800e-08  3.901713e-08 -1.212006e-08 -2.170200e-07 -1.525789e-06
## [36,]  1.180964e-08  3.563863e-08 -6.958137e-09 -2.394025e-08 -1.467711e-06
## [37,]  4.711290e-09  1.416982e-08  2.325194e-09  2.816994e-08 -6.802049e-07
## [38,] -3.130595e-07 -4.771561e-08  3.825772e-08 -1.063770e-06 -1.577589e-06
## [39,] -6.022644e-07 -3.206388e-08 -1.325704e-07 -1.152608e-06 -1.307899e-06
## [40,] -1.179901e-07 -5.098472e-07  8.263262e-08 -9.644929e-07 -1.631372e-06
## [41,] -1.281785e-07  8.838714e-08 -3.317733e-07 -9.968195e-08  2.001125e-07
## [42,] -2.169825e-07 -2.793952e-07  1.046587e-07 -3.255198e-06 -1.537808e-06
## [43,]  2.771580e-07  1.253527e-07  1.282627e-08 -4.419128e-06 -5.734921e-07
## [44,]  3.816294e-08  4.646769e-08 -1.531131e-09 -2.442973e-06  1.359063e-07
## [45,] -4.114458e-08 -3.526498e-08  1.801878e-08 -4.939579e-06  2.612450e-06
## [46,]  3.726954e-08 -2.142613e-09  1.423179e-08 -3.718307e-06  1.498786e-06
## [47,] -5.625748e-09  4.928918e-08 -9.863621e-09 -2.141715e-06 -5.134441e-07
## [48,] -4.042888e-08  4.160931e-08 -1.364946e-08 -1.685941e-06 -5.009010e-07
## [49,] -3.165965e-08  1.876286e-08 -1.443367e-09 -7.354414e-07 -2.441658e-07
## [50,] -4.032367e-07 -2.049082e-08  5.189167e-08 -2.509648e-06 -2.717195e-06
## [51,] -8.619167e-07 -1.276665e-07 -1.785099e-07 -2.035078e-06 -2.655996e-06
## [52,] -3.840850e-07 -5.534860e-07  1.192660e-08 -2.645357e-06 -2.160539e-06
## [53,] -1.328912e-07  4.022511e-08 -3.523298e-07  4.758640e-07  2.334192e-08
## [54,]  1.154484e-06  1.665468e-08  2.678160e-07  3.137169e-06 -4.029583e-07
## [55,]  5.755211e-07  4.311271e-07 -3.118368e-08 -2.834376e-07 -6.546455e-06
## [56,]  1.773287e-07  1.769495e-07 -2.491334e-08 -1.779394e-07 -4.985554e-06
## [57,] -1.756983e-07  2.381651e-07 -1.051070e-07  5.695182e-06 -8.866833e-06
## [58,] -1.657271e-07  1.501298e-07 -7.113805e-08  3.509972e-06 -5.812330e-06
## [59,] -7.666397e-08  1.031346e-07 -3.468609e-08 -7.035241e-07 -3.224205e-06
## [60,] -3.614804e-08  8.414379e-08 -2.399877e-08 -9.519382e-07 -2.567351e-06
## [61,] -3.438891e-08  3.893198e-08  1.232655e-09 -5.381691e-07 -1.279244e-06
## [62,] -9.055465e-07 -6.087623e-08  1.263225e-07 -4.557152e-06 -5.646722e-06
## [63,] -1.835181e-06 -2.615107e-07 -3.499150e-07 -4.848405e-06 -5.345635e-06
## [64,] -4.835596e-07 -1.002105e-06  1.484781e-07 -2.461889e-06 -4.270122e-06
## [65,] -4.107966e-07  1.344995e-07 -5.750123e-07 -3.900716e-07  3.011435e-07
## [66,] -1.998392e-05 -1.397648e-06 -2.320375e-06  5.701084e-05  8.019525e-05
##               [,16]         [,17]         [,18]         [,19]         [,20]
##  [1,] -5.381599e-06 -1.674575e-05 -5.750375e-07  6.691208e-07 -1.667181e-07
##  [2,]  3.676965e-07 -1.778038e-06 -6.388654e-08  1.101250e-07  8.788874e-09
##  [3,] -2.667658e-07 -1.544202e-06  2.977858e-08  2.470961e-07  4.976082e-10
##  [4,] -4.497462e-07 -1.242393e-06 -4.156419e-08  3.684964e-08 -1.320205e-07
##  [5,]  3.740820e-07 -2.595720e-06 -1.103108e-07  3.930172e-08 -7.651909e-08
##  [6,]  2.950458e-07 -1.698675e-06 -7.183237e-08  2.480451e-08 -8.415158e-09
##  [7,] -5.040960e-07 -8.595517e-07 -1.870980e-08  1.741412e-08  4.808187e-09
##  [8,] -6.186057e-07 -1.176875e-06 -5.463448e-08  5.040355e-08 -3.180114e-09
##  [9,] -3.995322e-07 -7.766387e-07 -4.328024e-08  4.201761e-08 -4.449448e-09
## [10,] -1.520142e-06 -1.597907e-06 -3.028541e-08  5.072285e-08  1.088968e-08
## [11,] -1.686009e-06 -1.716648e-06 -7.172254e-09  7.060838e-08  9.369027e-09
## [12,] -1.190856e-06 -1.243791e-06 -8.914043e-08  2.431847e-08  1.452754e-08
## [13,] -6.151977e-08  1.264394e-07  4.791025e-08  6.511705e-09 -7.452063e-10
## [14,] -3.379213e-05 -8.805917e-07 -1.161310e-06 -1.492399e-06 -9.849688e-07
## [15,] -6.456675e-06 -5.313877e-05 -3.585316e-07  2.140465e-07  5.645669e-10
## [16,]  2.688876e-04  1.189826e-05 -4.273354e-06 -4.495014e-06 -2.339052e-06
## [17,]  1.189826e-05  1.406581e-04  3.308027e-06 -7.500736e-07 -9.309850e-08
## [18,] -4.273354e-06  3.308027e-06  3.992500e-04 -3.970645e-07 -1.517052e-08
## [19,] -4.495014e-06 -7.500736e-07 -3.970645e-07  3.989165e-04 -2.853052e-07
## [20,] -2.339052e-06 -9.309850e-08 -1.517052e-08 -2.853052e-07  3.995278e-04
## [21,] -4.473218e-06  2.287149e-06  1.573681e-07  4.320544e-08 -4.162721e-07
## [22,] -3.196786e-06  9.363954e-07  8.620586e-08  6.333207e-08  1.067930e-08
## [23,] -1.959981e-06 -5.974657e-07  2.814093e-08  3.772485e-08  2.387650e-08
## [24,] -1.289203e-06 -1.774781e-07  5.388351e-08 -7.491040e-09  2.454314e-08
## [25,] -5.289950e-07 -5.101447e-08  2.966041e-08 -1.084643e-08  1.203962e-08
## [26,] -2.971668e-06 -2.719506e-06 -1.354465e-08  6.523846e-08  3.105292e-08
## [27,] -2.551408e-06 -3.080097e-06  9.329887e-08  7.159134e-08  2.495522e-08
## [28,] -2.919469e-06 -7.077973e-07 -2.491005e-07  2.358181e-09  3.884556e-08
## [29,]  5.771028e-07 -4.019215e-07  1.564363e-07  1.083594e-08 -7.434538e-09
## [30,]  5.807605e-06 -1.158864e-05  3.766871e-07  1.834637e-07 -4.703061e-08
## [31,] -1.547432e-06 -6.916594e-06 -1.887727e-08  3.016250e-08 -5.025529e-08
## [32,] -7.232619e-07 -3.181263e-06 -1.041035e-07 -5.148240e-08 -1.052328e-07
## [33,]  4.482534e-06 -6.556305e-06 -2.197204e-07 -1.092377e-07  5.680920e-09
## [34,]  2.445647e-06 -3.870918e-06 -1.209667e-07 -7.157145e-08 -1.907456e-08
## [35,] -1.174057e-06 -1.618519e-06  2.808723e-09 -6.496562e-09  1.997100e-08
## [36,] -1.347999e-06 -1.389200e-06 -4.544329e-09  1.425671e-08  1.657466e-08
## [37,] -8.008813e-07 -8.779780e-07 -1.090258e-08  1.753003e-08  7.469433e-09
## [38,] -5.712441e-06 -5.476238e-06 -3.321526e-08  1.166769e-07  5.848033e-08
## [39,] -6.248217e-06 -4.990945e-06  1.623669e-08  1.208736e-07  6.105675e-08
## [40,] -2.685375e-06 -4.529248e-06 -1.102905e-07  4.855430e-08  4.135475e-08
## [41,] -4.858983e-07  1.067289e-06  7.645460e-08 -2.906262e-09  1.784942e-09
## [42,] -2.538317e-05  2.662167e-05 -2.794567e-06 -1.363861e-06  4.346824e-08
## [43,] -1.991620e-05 -1.895548e-06 -1.454015e-06 -3.392569e-06 -7.543024e-07
## [44,] -8.154510e-06 -3.414253e-07 -6.292772e-08 -7.977661e-07 -1.110705e-06
## [45,] -1.634660e-05  9.752626e-06  5.716008e-07  2.223998e-07 -1.139175e-06
## [46,] -1.021998e-05  4.035744e-06  2.900836e-07  1.983088e-07  3.956704e-08
## [47,] -4.772521e-06 -1.614030e-06  7.873627e-08  6.409914e-08  6.053195e-08
## [48,] -3.287756e-06 -1.562192e-06  6.295119e-08  2.210313e-08  4.763843e-08
## [49,] -1.770460e-06 -1.095463e-06  2.352955e-08  1.594709e-08  2.405802e-08
## [50,] -1.183009e-05 -1.125426e-05 -6.451247e-08  2.442020e-07  1.214519e-07
## [51,] -1.030496e-05 -1.357657e-05  2.142936e-07  2.802819e-07  1.051331e-07
## [52,] -1.119531e-05 -9.018254e-07 -6.593573e-07  1.162043e-08  1.289639e-07
## [53,]  2.762244e-06 -2.495408e-06  4.053066e-07  4.095445e-08 -2.842925e-08
## [54,]  5.044793e-05 -8.426471e-05  4.826079e-06  1.025338e-06 -5.466062e-07
## [55,] -6.278792e-06 -3.788615e-05  1.823378e-07 -1.514062e-06 -6.219568e-07
## [56,] -2.674165e-06 -1.194133e-05 -4.987145e-07 -6.541308e-07 -5.997125e-08
## [57,]  2.142850e-05 -2.470521e-05 -8.684295e-07 -5.301257e-07  4.309992e-07
## [58,]  1.051319e-05 -1.359908e-05 -4.247013e-07 -3.245104e-07 -5.912401e-08
## [59,] -3.900965e-06 -4.631592e-06  2.689912e-08 -1.628804e-08  6.387016e-08
## [60,] -4.281018e-06 -3.365966e-06  4.265312e-08  2.008900e-08  6.043807e-08
## [61,] -2.816439e-06 -2.423329e-06  7.060323e-09  3.427589e-08  3.434482e-08
## [62,] -2.404866e-05 -2.317753e-05 -1.407543e-07  4.934026e-07  2.454825e-07
## [63,] -2.820111e-05 -1.903181e-05 -2.593597e-07  4.421804e-07  2.795615e-07
## [64,] -4.957648e-06 -2.326876e-05  3.029971e-07  3.388137e-07  9.307792e-08
## [65,] -4.255576e-06  7.821308e-06 -1.505735e-07 -1.051859e-07  3.258090e-08
## [66,]  9.052213e-05  1.570192e-04  1.017970e-05 -1.238721e-05  1.297193e-06
##               [,21]         [,22]         [,23]         [,24]         [,25]
##  [1,] -1.576359e-07  2.135272e-07 -1.467427e-07  1.796292e-07  2.113376e-07
##  [2,] -4.657183e-08 -2.217752e-08  6.153365e-09 -8.322053e-09 -6.420554e-09
##  [3,] -4.760074e-08 -1.357117e-08  7.647601e-09 -1.707008e-08 -1.238423e-08
##  [4,] -1.065255e-07 -7.481108e-09  5.037906e-09 -4.730141e-08 -3.168123e-08
##  [5,]  6.655278e-08  1.454594e-07  1.330210e-08 -8.929771e-08 -5.813182e-08
##  [6,]  1.106016e-07  1.734109e-07 -3.273093e-08 -7.154553e-08 -3.400915e-08
##  [7,] -9.773000e-09 -4.358655e-08 -1.737268e-07 -5.151375e-08  7.635810e-11
##  [8,] -4.770929e-08 -2.322016e-08 -4.131356e-08  3.156510e-07  1.599008e-07
##  [9,] -3.686993e-08 -2.404004e-10  5.624809e-09  1.411342e-07  1.585759e-07
## [10,] -1.901321e-08  6.596301e-09  1.587341e-08 -2.242782e-08  2.066964e-08
## [11,] -2.483240e-08  1.242561e-08  1.404120e-08 -5.485238e-08 -3.564940e-08
## [12,] -3.007519e-09  3.620781e-09  1.926995e-08  9.563913e-09  3.818931e-09
## [13,]  1.541119e-09  3.570675e-09 -1.462228e-09 -8.460010e-09 -5.153329e-09
## [14,] -1.760698e-06 -1.607377e-06 -1.256562e-06 -1.253758e-06 -5.729760e-07
## [15,]  6.145698e-07  4.636755e-07 -2.052442e-07  3.236918e-07  2.776068e-07
## [16,] -4.473218e-06 -3.196786e-06 -1.959981e-06 -1.289203e-06 -5.289950e-07
## [17,]  2.287149e-06  9.363954e-07 -5.974657e-07 -1.774781e-07 -5.101447e-08
## [18,]  1.573681e-07  8.620586e-08  2.814093e-08  5.388351e-08  2.966041e-08
## [19,]  4.320544e-08  6.333207e-08  3.772485e-08 -7.491040e-09 -1.084643e-08
## [20,] -4.162721e-07  1.067930e-08  2.387650e-08  2.454314e-08  1.203962e-08
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## [25,]  2.918742e-08  6.888680e-09 -8.719927e-09 -3.079110e-07  3.997424e-04
## [26,]  6.356204e-09  1.831286e-08  3.929707e-08  1.677693e-08 -1.419907e-07
## [27,] -1.385293e-08  8.338002e-09  3.492448e-08  1.167131e-08 -1.481140e-08
## [28,]  6.551929e-08  3.040045e-08  4.014348e-08  8.438347e-08  4.340656e-08
## [29,] -2.400916e-08 -1.106652e-08 -5.215098e-09 -1.636788e-08 -5.743865e-09
## [30,] -3.283011e-07 -1.789873e-07  6.767352e-09 -8.889861e-09 -4.837008e-09
## [31,] -1.202654e-07 -4.875052e-08  4.577485e-08  3.250078e-08  1.475350e-08
## [32,] -3.686544e-08 -3.984315e-08  2.151647e-08  5.187452e-09  1.058034e-10
## [33,]  5.846461e-07  4.600201e-07  3.645556e-08  6.774709e-08  4.341578e-08
## [34,]  5.066023e-07  4.563735e-07 -1.120902e-07  2.260961e-08  3.881797e-08
## [35,]  1.881500e-08 -1.459194e-07 -3.562454e-07 -3.986560e-08  3.680993e-08
## [36,] -2.214930e-09 -4.216020e-08 -8.889702e-08 -1.446716e-07  3.849369e-08
## [37,] -6.262487e-09  1.411177e-09  6.122364e-09  2.366155e-08  4.911803e-08
## [38,]  3.475013e-09  2.652187e-08  7.552528e-08  5.669662e-08 -9.729560e-08
## [39,]  1.954770e-08  4.015152e-08  7.527859e-08  4.088950e-08 -2.888518e-09
## [40,] -8.722976e-09 -1.153581e-08  5.865312e-08  1.105005e-07  5.820959e-08
## [41,]  2.550457e-08  1.788072e-08 -2.904708e-09 -1.352358e-08 -3.524430e-09
## [42,]  1.023396e-06  5.602787e-07  1.128830e-07  2.286727e-07  1.246745e-07
## [43,]  2.591966e-07  2.761953e-07  1.571783e-07 -3.064501e-08 -4.517390e-08
## [44,] -1.138540e-06  5.636505e-08  7.768005e-08  7.240014e-08  3.455634e-08
## [45,] -3.808441e-06 -1.281385e-06  2.902285e-09  1.035367e-07  5.277186e-08
## [46,] -1.351679e-06 -1.952804e-06 -1.091196e-06 -1.025890e-07  1.007288e-08
## [47,] -1.586854e-08 -1.114577e-06 -1.411549e-06 -2.544525e-07  2.519372e-08
## [48,]  6.488392e-08 -1.073692e-07 -3.059675e-07 -1.451946e-06 -4.157482e-07
## [49,]  2.983599e-08  1.423586e-08 -4.391350e-09 -4.284246e-07 -2.531741e-07
## [50,]  1.314663e-08  5.931862e-08  1.557805e-07  1.079229e-07 -2.690149e-07
## [51,] -6.675680e-08  1.859184e-08  1.496162e-07  8.613689e-08 -6.942056e-09
## [52,]  2.463119e-07  1.313031e-07  1.250878e-07  2.255376e-07  1.152698e-07
## [53,] -1.092586e-07 -5.740604e-08 -1.603083e-08 -3.921166e-08 -1.161996e-08
## [54,] -2.596819e-06 -1.381563e-06 -6.245206e-08 -3.324395e-07 -1.895262e-07
## [55,] -6.547574e-07 -2.594616e-07  2.164174e-07  5.192587e-08  5.233199e-09
## [56,]  3.332958e-07 -1.207692e-07  7.876975e-08  3.409447e-08  1.057469e-08
## [57,]  3.223392e-06  1.863611e-06  8.071581e-08  3.376918e-07  2.228410e-07
## [58,]  2.162261e-06  1.537983e-06 -4.691899e-07  1.719166e-07  1.839774e-07
## [59,]  8.548765e-08 -5.458938e-07 -1.079924e-06 -9.736516e-08  1.036046e-07
## [60,]  4.714146e-08 -1.084068e-07 -2.511644e-07 -1.161458e-06 -2.144264e-07
## [61,]  1.956972e-08  1.261988e-08  1.396718e-08 -2.387686e-07 -1.274245e-07
## [62,]  1.759972e-08  1.144554e-07  3.163913e-07  2.278124e-07 -4.364556e-07
## [63,]  1.741887e-07  2.064756e-07  3.302886e-07  2.360548e-07  3.536161e-08
## [64,] -2.830533e-07 -1.538591e-07  1.830531e-07  2.777176e-07  1.376160e-07
## [65,]  2.265740e-07  1.261473e-07 -1.008803e-09  3.991976e-09  2.015816e-08
## [66,]  7.913723e-06 -1.554536e-06  9.969276e-07  1.880251e-05  1.202901e-05
##               [,26]         [,27]         [,28]         [,29]         [,30]
##  [1,]  2.305971e-07  1.416995e-07 -8.428031e-07  6.629177e-08 -2.068694e-08
##  [2,]  3.659493e-08  7.252342e-08 -6.210066e-08  4.842842e-08 -6.364691e-07
##  [3,]  3.794353e-08  4.696737e-08 -1.041957e-09  1.018248e-08 -1.689326e-07
##  [4,]  3.333505e-08  4.327278e-08 -2.613757e-08  1.242769e-08  7.498691e-08
##  [5,]  5.261776e-08  7.285785e-08 -5.768223e-08  2.509570e-08  2.557167e-07
##  [6,]  3.557977e-08  4.845908e-08 -3.076281e-08  1.558277e-08  1.755555e-07
##  [7,]  2.886154e-08  3.255468e-08  7.841754e-09  3.937542e-09  8.259817e-08
##  [8,]  4.143373e-08  4.749011e-08 -4.677624e-08  1.675159e-08  1.043146e-07
##  [9,]  6.363559e-08  3.421820e-08 -4.647118e-08  1.419238e-08  6.302210e-08
## [10,] -5.299725e-08 -6.824987e-08 -4.843687e-08  1.731877e-08  1.144692e-07
## [11,] -7.381944e-08 -2.595183e-07 -8.238001e-08 -3.672564e-08  2.172762e-07
## [12,] -1.410182e-09 -9.924655e-09 -2.452305e-07  1.560409e-09 -1.628832e-07
## [13,]  6.167946e-09 -5.159390e-08 -8.279749e-09 -2.167409e-07  1.373135e-07
## [14,] -9.838415e-07 -6.017925e-07 -1.196349e-06  2.077234e-07  1.847011e-07
## [15,] -6.433824e-07 -6.803010e-07 -7.070486e-07 -5.367791e-08 -8.150956e-07
## [16,] -2.971668e-06 -2.551408e-06 -2.919469e-06  5.771028e-07  5.807605e-06
## [17,] -2.719506e-06 -3.080097e-06 -7.077973e-07 -4.019215e-07 -1.158864e-05
## [18,] -1.354465e-08  9.329887e-08 -2.491005e-07  1.564363e-07  3.766871e-07
## [19,]  6.523846e-08  7.159134e-08  2.358181e-09  1.083594e-08  1.834637e-07
## [20,]  3.105292e-08  2.495522e-08  3.884556e-08 -7.434538e-09 -4.703061e-08
## [21,]  6.356204e-09 -1.385293e-08  6.551929e-08 -2.400916e-08 -3.283011e-07
## [22,]  1.831286e-08  8.338002e-09  3.040045e-08 -1.106652e-08 -1.789873e-07
## [23,]  3.929707e-08  3.492448e-08  4.014348e-08 -5.215098e-09  6.767352e-09
## [24,]  1.677693e-08  1.167131e-08  8.438347e-08 -1.636788e-08 -8.889861e-09
## [25,] -1.419907e-07 -1.481140e-08  4.340656e-08 -5.743865e-09 -4.837008e-09
## [26,]  3.993382e-04 -3.243390e-07 -8.871558e-08  4.690930e-08  1.772339e-07
## [27,] -3.243390e-07  3.993816e-04 -7.429710e-08 -1.265836e-07  2.348146e-07
## [28,] -8.871558e-08 -7.429710e-08  3.995015e-04  9.082251e-08 -2.016037e-08
## [29,]  4.690930e-08 -1.265836e-07  9.082251e-08  3.996654e-04  6.976405e-08
## [30,]  1.772339e-07  2.348146e-07 -2.016037e-08  6.976405e-08  3.976542e-04
## [31,]  1.672299e-07  1.756201e-07  1.015284e-07  5.087631e-09 -5.532627e-07
## [32,]  8.101355e-08  9.137120e-08  2.964819e-08  1.149149e-08  2.006360e-07
## [33,]  1.007769e-07  1.198486e-07  7.565515e-08  1.825785e-08  7.849665e-07
## [34,]  6.511219e-08  7.420824e-08  6.776749e-08  7.180697e-09  4.711641e-07
## [35,]  5.813683e-08  5.361554e-08  8.830609e-08 -8.040387e-09  1.585782e-07
## [36,]  5.741802e-08  5.203898e-08  5.370290e-08 -2.412603e-09  1.243522e-07
## [37,] -9.029015e-08  1.307418e-08  1.056043e-08  6.770048e-09  7.097700e-08
## [38,] -8.563156e-07 -4.297041e-07 -9.724351e-08  6.825754e-08  3.758501e-07
## [39,] -4.398771e-07 -9.275960e-07 -2.362262e-07 -1.602789e-07  6.293532e-07
## [40,] -7.981765e-08 -2.112354e-07 -4.843871e-07  1.729481e-08 -3.858961e-07
## [41,]  4.647599e-08 -1.846693e-07  8.153674e-09 -3.504054e-07  3.380826e-07
## [42,] -2.419825e-07  1.546752e-08 -7.050031e-07  3.964867e-07  4.582853e-06
## [43,]  2.610369e-07  2.995736e-07 -1.714716e-08  6.123378e-08  1.269205e-06
## [44,]  1.035941e-07  8.529314e-08  1.191817e-07 -2.235366e-08 -2.248376e-07
## [45,] -1.709227e-08 -8.346404e-08  1.508528e-07 -7.717737e-08 -1.350155e-06
## [46,]  3.146812e-08  8.267130e-11  7.240050e-08 -3.509818e-08 -6.661479e-07
## [47,]  9.848971e-08  8.366984e-08  1.359680e-07 -1.945446e-08  4.443899e-08
## [48,]  8.034363e-08  6.640226e-08  1.411659e-07 -2.061800e-08  8.423570e-08
## [49,] -2.727859e-07 -3.694069e-09  5.755302e-08  2.476530e-09  6.611741e-08
## [50,] -1.944505e-06 -9.648042e-07 -2.224876e-07  1.600433e-07  7.577968e-07
## [51,] -9.430926e-07 -1.840999e-06 -3.925516e-07 -3.272395e-07  6.855576e-07
## [52,] -3.023790e-07 -4.929527e-07 -1.086817e-06  1.901244e-07  6.322135e-07
## [53,]  1.665966e-07 -3.217512e-07  1.955203e-07 -5.903691e-07 -1.602470e-07
## [54,]  1.151382e-06  1.101964e-06  7.521414e-07 -1.573108e-07 -1.500402e-05
## [55,]  8.686536e-07  8.641482e-07  5.590533e-07 -3.994475e-08 -3.334986e-06
## [56,]  2.971640e-07  3.338230e-07  1.151858e-07  4.061419e-08  7.298997e-07
## [57,]  3.226996e-07  3.894950e-07  3.152214e-07  6.082253e-08  3.066483e-06
## [58,]  2.005198e-07  2.249367e-07  2.740641e-07  1.532146e-08  1.685436e-06
## [59,]  1.672678e-07  1.500677e-07  2.678429e-07 -2.796424e-08  4.218915e-07
## [60,]  1.460785e-07  1.257839e-07  2.056810e-07 -2.247028e-08  2.707661e-07
## [61,] -4.547424e-07  6.243348e-09  7.689765e-08  1.337061e-08  1.797335e-07
## [62,] -3.639177e-06 -1.855682e-06 -4.105976e-07  3.241082e-07  1.572402e-06
## [63,] -1.908531e-06 -3.519433e-06 -1.079069e-06 -4.183381e-07  2.802624e-06
## [64,] -3.526316e-07 -1.014600e-06 -1.219440e-06  1.316605e-08 -2.063006e-06
## [65,]  1.772738e-07 -5.789598e-07 -3.266018e-08 -7.313694e-07  1.615590e-06
## [66,] -3.574090e-06 -6.212121e-06  1.501326e-05 -3.646391e-06 -8.109113e-06
##               [,31]         [,32]         [,33]         [,34]         [,35]
##  [1,] -6.934362e-07 -6.556152e-07 -1.680073e-06 -1.648618e-06 -1.489113e-06
##  [2,] -2.509855e-07  1.371652e-08  1.197399e-07  7.303616e-08  2.992818e-08
##  [3,] -9.715332e-07 -2.455642e-07  7.260871e-08  5.210023e-08  2.126823e-08
##  [4,] -2.225146e-07 -4.316792e-07 -4.056919e-07 -1.049083e-08 -1.251886e-08
##  [5,]  1.284143e-07 -3.438460e-07 -1.265041e-06 -4.872420e-07 -5.861654e-08
##  [6,]  9.985607e-08  4.576997e-08 -4.400859e-07 -8.387073e-07 -5.307712e-07
##  [7,]  6.482915e-08  3.702257e-08  2.057449e-08 -4.895062e-07 -7.173043e-07
##  [8,]  7.050605e-08  4.912109e-08 -8.131006e-09 -8.284154e-08 -2.813866e-07
##  [9,]  3.883455e-08  3.105309e-08 -3.027241e-08 -3.213951e-08 -5.765654e-08
## [10,]  9.726996e-08  5.466462e-08  3.106650e-08  1.348251e-08  6.769729e-09
## [11,]  9.170936e-08  5.261459e-08 -1.077104e-08 -2.093552e-08 -2.034800e-08
## [12,]  8.584167e-08  5.004448e-08  7.234099e-08  4.963683e-08  3.901713e-08
## [13,] -8.496671e-09 -6.533962e-09 -2.455746e-08 -1.812585e-08 -1.212006e-08
## [14,]  3.319922e-07 -1.234361e-07  1.134604e-06  9.067545e-07 -2.170200e-07
## [15,] -2.023334e-06 -1.640479e-06 -2.840721e-06 -2.254085e-06 -1.525789e-06
## [16,] -1.547432e-06 -7.232619e-07  4.482534e-06  2.445647e-06 -1.174057e-06
## [17,] -6.916594e-06 -3.181263e-06 -6.556305e-06 -3.870918e-06 -1.618519e-06
## [18,] -1.887727e-08 -1.041035e-07 -2.197204e-07 -1.209667e-07  2.808723e-09
## [19,]  3.016250e-08 -5.148240e-08 -1.092377e-07 -7.157145e-08 -6.496562e-09
## [20,] -5.025529e-08 -1.052328e-07  5.680920e-09 -1.907456e-08  1.997100e-08
## [21,] -1.202654e-07 -3.686544e-08  5.846461e-07  5.066023e-07  1.881500e-08
## [22,] -4.875052e-08 -3.984315e-08  4.600201e-07  4.563735e-07 -1.459194e-07
## [23,]  4.577485e-08  2.151647e-08  3.645556e-08 -1.120902e-07 -3.562454e-07
## [24,]  3.250078e-08  5.187452e-09  6.774709e-08  2.260961e-08 -3.986560e-08
## [25,]  1.475350e-08  1.058034e-10  4.341578e-08  3.881797e-08  3.680993e-08
## [26,]  1.672299e-07  8.101355e-08  1.007769e-07  6.511219e-08  5.813683e-08
## [27,]  1.756201e-07  9.137120e-08  1.198486e-07  7.420824e-08  5.361554e-08
## [28,]  1.015284e-07  2.964819e-08  7.565515e-08  6.776749e-08  8.830609e-08
## [29,]  5.087631e-09  1.149149e-08  1.825785e-08  7.180697e-09 -8.040387e-09
## [30,] -5.532627e-07  2.006360e-07  7.849665e-07  4.711641e-07  1.585782e-07
## [31,]  3.970439e-04 -6.274196e-07  4.072655e-07  2.653300e-07  1.398439e-07
## [32,] -6.274196e-07  3.989709e-04 -9.872829e-07  9.493283e-08  6.484070e-08
## [33,]  4.072655e-07 -9.872829e-07  3.966238e-04 -1.094630e-06  1.252737e-07
## [34,]  2.653300e-07  9.493283e-08 -1.094630e-06  3.982033e-04 -9.740398e-07
## [35,]  1.398439e-07  6.484070e-08  1.252737e-07 -9.740398e-07  3.986554e-04
## [36,]  1.148684e-07  5.541028e-08  1.151254e-07 -4.380649e-08 -2.886213e-07
## [37,]  6.177324e-08  3.159441e-08  4.437430e-08  3.085489e-08 -1.026476e-08
## [38,]  3.467484e-07  1.674633e-07  2.329397e-07  1.535628e-07  1.281750e-07
## [39,]  3.299752e-07  1.473733e-07  1.710518e-07  1.116587e-07  1.073141e-07
## [40,]  2.701866e-07  1.530991e-07  3.414513e-07  2.381502e-07  1.728864e-07
## [41,] -3.979234e-08 -3.606554e-08 -9.245353e-08 -6.049719e-08 -2.913861e-08
## [42,] -7.093730e-07 -9.417251e-07 -1.769878e-06 -1.020197e-06 -1.538001e-07
## [43,] -1.576309e-06 -6.201270e-07 -4.580305e-07 -3.202783e-07 -2.376187e-08
## [44,] -6.143412e-07 -7.227421e-08  4.341632e-07 -6.771821e-08  5.608079e-08
## [45,] -5.364341e-07  2.347454e-07  3.007795e-06  1.954273e-06 -6.316658e-08
## [46,] -2.174373e-07 -1.525624e-07  1.949270e-06  1.544081e-06 -6.048024e-07
## [47,]  1.362750e-07  5.696726e-08  1.907992e-07 -3.972538e-07 -1.050251e-06
## [48,]  1.363710e-07  5.400470e-08  1.515102e-07  1.082126e-08 -1.942779e-07
## [49,]  8.386901e-08  3.472416e-08  8.397713e-08  6.710632e-08  3.596127e-08
## [50,]  7.015008e-07  3.373029e-07  4.610500e-07  3.024373e-07  2.562936e-07
## [51,]  7.562487e-07  4.055767e-07  5.983249e-07  3.800437e-07  2.676613e-07
## [52,]  3.215301e-07  3.998505e-08  6.976673e-08  1.003800e-07  2.229780e-07
## [53,]  3.162457e-08  7.584082e-08  1.577699e-07  8.609202e-08  2.555938e-10
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## [59,]  3.868689e-07  1.737771e-07  3.340118e-07 -2.427022e-06 -3.276068e-06
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## [62,]  1.446178e-06  6.951382e-07  9.622635e-07  6.313910e-07  5.283783e-07
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## [66,] -3.141515e-06 -4.360917e-06  1.755875e-05  1.581941e-05  1.449813e-05
##               [,36]         [,37]         [,38]         [,39]         [,40]
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##  [4,]  1.060371e-08  1.694841e-08  6.500020e-08  6.637148e-08  5.765748e-09
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##  [7,] -2.293525e-07 -6.138817e-09  6.055110e-08  5.704087e-08  3.951713e-08
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## [23,] -8.889702e-08  6.122364e-09  7.552528e-08  7.527859e-08  5.865312e-08
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## [25,]  3.849369e-08  4.911803e-08 -9.729560e-08 -2.888518e-09  5.820959e-08
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## [27,]  5.203898e-08  1.307418e-08 -4.297041e-07 -9.275960e-07 -2.112354e-07
## [28,]  5.370290e-08  1.056043e-08 -9.724351e-08 -2.362262e-07 -4.843871e-07
## [29,] -2.412603e-09  6.770048e-09  6.825754e-08 -1.602789e-07  1.729481e-08
## [30,]  1.243522e-07  7.097700e-08  3.758501e-07  6.293532e-07 -3.858961e-07
## [31,]  1.148684e-07  6.177324e-08  3.467484e-07  3.299752e-07  2.701866e-07
## [32,]  5.541028e-08  3.159441e-08  1.674633e-07  1.473733e-07  1.530991e-07
## [33,]  1.151254e-07  4.437430e-08  2.329397e-07  1.710518e-07  3.414513e-07
## [34,] -4.380649e-08  3.085489e-08  1.535628e-07  1.116587e-07  2.381502e-07
## [35,] -2.886213e-07 -1.026476e-08  1.281750e-07  1.073141e-07  1.728864e-07
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## [40,]  1.165382e-07  3.465388e-08 -1.781969e-07 -3.817525e-07  3.989885e-04
## [41,] -1.844914e-08  3.941138e-09  1.124840e-07 -3.711970e-07  1.588797e-07
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## [43,]  6.424889e-08  7.359097e-08  4.711540e-07  4.287430e-07  3.401782e-07
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## [47,] -2.512491e-07 -3.014952e-09  1.973376e-07  1.865109e-07  2.039567e-07
## [48,] -1.243999e-06 -2.845004e-07  1.751967e-07  1.499560e-07  2.207371e-07
## [49,] -2.656853e-07 -1.510042e-07 -4.435139e-07  1.157722e-08  9.688268e-08
## [50,]  2.319485e-07 -4.118895e-07 -3.664743e-06 -1.929948e-06 -3.286002e-07
## [51,]  2.382349e-07  6.442480e-08 -1.837701e-06 -3.495743e-06 -9.532017e-07
## [52,]  1.427327e-07  2.573549e-08 -5.782582e-07 -1.266774e-06 -1.310331e-06
## [53,]  6.715513e-09  2.994976e-08  3.386214e-07 -4.090963e-07  3.495864e-08
## [54,]  7.230974e-07  4.697526e-07  2.425646e-06  3.603348e-06 -1.548005e-06
## [55,]  5.293449e-07  3.143642e-07  1.773053e-06  1.808921e-06  1.009766e-06
## [56,]  2.000907e-07  1.119853e-07  6.127218e-07  5.357647e-07  5.772158e-07
## [57,]  4.392825e-07  1.426035e-07  7.787123e-07  5.183147e-07  1.350634e-06
## [58,] -1.939002e-08  8.983962e-08  4.882983e-07  3.265565e-07  8.865828e-07
## [59,] -5.722497e-07 -1.566573e-08  3.652780e-07  3.066522e-07  5.035681e-07
## [60,] -3.324404e-06 -9.826514e-07  3.055832e-07  2.612635e-07  3.725225e-07
## [61,] -9.545774e-07 -5.713729e-07 -8.970551e-07  1.930788e-08  1.550770e-07
## [62,]  4.708211e-07 -8.099735e-07 -7.532276e-06 -4.122888e-06 -7.109563e-07
## [63,]  4.264268e-07  1.020723e-07 -4.140365e-06 -7.425711e-06 -1.736764e-06
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## [65,] -8.137951e-08 -8.717895e-09  3.991930e-07 -1.171072e-06  3.912976e-07
## [66,]  4.450066e-06 -2.015044e-06 -4.769992e-06 -8.082144e-06  1.737034e-05
##               [,41]         [,42]         [,43]         [,44]         [,45]
##  [1,]  2.122005e-07 -3.385838e-06  1.596640e-06 -6.452458e-08 -3.836549e-08
##  [2,]  1.470532e-07  2.224558e-07  2.573600e-07 -3.003364e-09 -1.882791e-07
##  [3,] -9.390409e-09 -3.590062e-07  6.565460e-08 -5.889629e-08 -1.554138e-07
##  [4,] -3.158401e-09 -3.513994e-07  8.925257e-08 -1.107412e-07 -9.362926e-08
##  [5,] -8.084341e-09 -7.247000e-07  1.813037e-07 -6.305295e-09  4.525258e-07
##  [6,] -7.937527e-09 -4.829355e-07  9.467207e-08 -3.096916e-08  3.854712e-07
##  [7,] -6.724786e-09 -1.604540e-07  7.353569e-08  1.471957e-08 -6.172124e-08
##  [8,]  3.538177e-09 -3.245248e-07  2.116831e-07 -3.656012e-09 -1.075114e-07
##  [9,]  1.028622e-08 -2.370370e-07  1.751692e-07 -8.063586e-09 -6.924746e-08
## [10,]  4.032289e-08 -2.383910e-07  2.094606e-07  3.966682e-08 -5.031207e-08
## [11,] -1.281785e-07 -2.169825e-07  2.771580e-07  3.816294e-08 -4.114458e-08
## [12,]  8.838714e-08 -2.793952e-07  1.253527e-07  4.646769e-08 -3.526498e-08
## [13,] -3.317733e-07  1.046587e-07  1.282627e-08 -1.531131e-09  1.801878e-08
## [14,] -9.968195e-08 -3.255198e-06 -4.419128e-06 -2.442973e-06 -4.939579e-06
## [15,]  2.001125e-07 -1.537808e-06 -5.734921e-07  1.359063e-07  2.612450e-06
## [16,] -4.858983e-07 -2.538317e-05 -1.991620e-05 -8.154510e-06 -1.634660e-05
## [17,]  1.067289e-06  2.662167e-05 -1.895548e-06 -3.414253e-07  9.752626e-06
## [18,]  7.645460e-08 -2.794567e-06 -1.454015e-06 -6.292772e-08  5.716008e-07
## [19,] -2.906262e-09 -1.363861e-06 -3.392569e-06 -7.977661e-07  2.223998e-07
## [20,]  1.784942e-09  4.346824e-08 -7.543024e-07 -1.110705e-06 -1.139175e-06
## [21,]  2.550457e-08  1.023396e-06  2.591966e-07 -1.138540e-06 -3.808441e-06
## [22,]  1.788072e-08  5.602787e-07  2.761953e-07  5.636505e-08 -1.281385e-06
## [23,] -2.904708e-09  1.128830e-07  1.571783e-07  7.768005e-08  2.902285e-09
## [24,] -1.352358e-08  2.286727e-07 -3.064501e-08  7.240014e-08  1.035367e-07
## [25,] -3.524430e-09  1.246745e-07 -4.517390e-08  3.455634e-08  5.277186e-08
## [26,]  4.647599e-08 -2.419825e-07  2.610369e-07  1.035941e-07 -1.709227e-08
## [27,] -1.846693e-07  1.546752e-08  2.995736e-07  8.529314e-08 -8.346404e-08
## [28,]  8.153674e-09 -7.050031e-07 -1.714716e-08  1.191817e-07  1.508528e-07
## [29,] -3.504054e-07  3.964867e-07  6.123378e-08 -2.235366e-08 -7.717737e-08
## [30,]  3.380826e-07  4.582853e-06  1.269205e-06 -2.248376e-07 -1.350155e-06
## [31,] -3.979234e-08 -7.093730e-07 -1.576309e-06 -6.143412e-07 -5.364341e-07
## [32,] -3.606554e-08 -9.417251e-07 -6.201270e-07 -7.227421e-08  2.347454e-07
## [33,] -9.245353e-08 -1.769878e-06 -4.580305e-07  4.341632e-07  3.007795e-06
## [34,] -6.049719e-08 -1.020197e-06 -3.202783e-07 -6.771821e-08  1.954273e-06
## [35,] -2.913861e-08 -1.538001e-07 -2.376187e-08  5.608079e-08 -6.316658e-08
## [36,] -1.844914e-08 -1.433155e-07  6.424889e-08  5.084019e-08 -6.092605e-08
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## [38,]  1.124840e-07 -5.270361e-07  4.711540e-07  1.945076e-07 -7.614165e-08
## [39,] -3.711970e-07 -6.223984e-07  4.287430e-07  2.030156e-07  4.801616e-09
## [40,]  1.588797e-07  3.199326e-08  3.401782e-07  1.316978e-07 -1.953315e-07
## [41,]  3.994185e-04 -8.508022e-08 -9.344863e-08  4.614208e-09  1.232485e-07
## [42,] -8.508022e-08  3.822952e-04 -7.592981e-06 -7.817364e-08  3.892717e-06
## [43,] -9.344863e-08 -7.592981e-06  3.864744e-04 -2.905182e-06  1.102330e-06
## [44,]  4.614208e-09 -7.817364e-08 -2.905182e-06  3.965618e-04 -3.744683e-06
## [45,]  1.232485e-07  3.892717e-06  1.102330e-06 -3.744683e-06  3.862773e-04
## [46,]  6.866381e-08  1.961646e-06  8.794519e-07  1.456307e-07 -4.966903e-06
## [47,] -1.899651e-08  2.887198e-07  2.724303e-07  1.931622e-07 -8.673818e-08
## [48,] -2.800867e-08  1.733209e-07  9.418422e-08  1.464040e-07  1.055975e-07
## [49,]  4.675223e-09  2.804180e-08  6.551571e-08  7.494927e-08  4.263591e-08
## [50,]  2.275336e-07 -1.056486e-06  9.813770e-07  4.048152e-07 -1.325775e-07
## [51,] -5.260970e-07  2.444701e-07  1.278223e-06  3.585210e-07 -4.317852e-07
## [52,] -1.243038e-08 -3.559390e-06 -3.148425e-07  3.971234e-07  6.924829e-07
## [53,] -7.357028e-07  1.947036e-06  3.816061e-07 -8.377871e-08 -3.969912e-07
## [54,]  1.501211e-06  4.042948e-05  9.833220e-06 -1.927098e-06 -1.029014e-05
## [55,] -4.671475e-08  6.814852e-07 -1.042498e-05 -3.788115e-06 -2.829246e-06
## [56,] -1.380356e-07 -3.924654e-06 -3.970126e-06  4.335252e-07  2.150008e-06
## [57,] -3.781089e-07 -6.988689e-06 -2.288495e-06  2.667117e-06  1.511396e-05
## [58,] -2.287855e-07 -3.636853e-06 -1.444354e-06 -2.027422e-07  8.684625e-06
## [59,] -8.280607e-08 -3.322107e-07 -6.274099e-08  1.835974e-07 -4.114046e-08
## [60,] -5.568852e-08 -9.998814e-08  9.288262e-08  1.832587e-07 -3.928512e-08
## [61,]  1.122687e-08 -1.553834e-07  1.427460e-07  1.083718e-07 -1.967593e-08
## [62,]  5.097315e-07 -2.219269e-06  1.983805e-06  8.186536e-07 -3.068923e-07
## [63,] -1.052652e-06 -4.324336e-06  1.346999e-06  9.167408e-07  3.165582e-07
## [64,]  4.232365e-07  4.208172e-06  2.411112e-06  3.258134e-07 -1.576663e-06
## [65,] -1.455154e-06 -2.775708e-06 -1.030571e-06  8.561463e-08  9.299378e-07
## [66,] -3.047119e-06  5.023227e-05 -4.959046e-05  2.045903e-06  9.203763e-06
##               [,46]         [,47]         [,48]         [,49]         [,50]
##  [1,]  5.649393e-07 -5.661797e-07 -4.665317e-07 -9.887992e-08 -7.369209e-08
##  [2,] -8.476549e-08  1.431463e-08  1.326987e-08  1.077033e-08  1.558924e-07
##  [3,] -5.263209e-08  1.170743e-08  3.609210e-09  6.296524e-09  1.592102e-07
##  [4,] -3.404431e-08 -1.762456e-08 -3.951104e-08 -1.425080e-08  1.286692e-07
##  [5,]  5.003071e-07 -1.592168e-08 -8.094951e-08 -3.045701e-08  2.047467e-07
##  [6,]  4.610482e-07 -1.400024e-07 -8.002421e-08 -1.440463e-08  1.419161e-07
##  [7,] -1.600412e-07 -3.627030e-07 -9.990984e-08  2.731475e-09  1.190931e-07
##  [8,] -4.380679e-08 -1.513211e-07 -2.459550e-07 -6.941742e-09  1.341218e-07
##  [9,]  7.612283e-09 -5.047498e-08 -5.083184e-08  1.895353e-08 -4.287190e-08
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## [48,] -2.875221e-07 -5.812871e-07  3.967164e-04 -9.362848e-07  3.524316e-07
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## [50,]  8.000912e-08  4.054323e-07  3.524316e-07 -8.771988e-07  3.924436e-04
## [51,] -6.341564e-08  3.859161e-07  3.313809e-07  5.492312e-08 -3.956906e-06
## [52,]  3.656655e-07  3.961479e-07  3.859590e-07  1.530260e-07 -1.235237e-06
## [53,] -1.958629e-07 -4.908911e-08 -4.169099e-08  2.392137e-08  7.373083e-07
## [54,] -5.038421e-06 -8.665338e-08  1.723785e-07  2.698182e-07  4.900000e-06
## [55,] -1.140386e-06  5.759170e-07  5.400821e-07  3.608912e-07  3.602323e-06
## [56,] -4.520036e-07  2.185525e-07  2.135529e-07  1.350443e-07  1.238242e-06
## [57,]  8.285830e-06  6.829239e-07  6.418158e-07  3.451654e-07  1.537019e-06
## [58,]  5.937705e-06 -1.500440e-06  1.371251e-07  2.638460e-07  9.636324e-07
## [59,] -2.130388e-06 -3.470666e-06 -5.903624e-07  9.952836e-08  7.356637e-07
## [60,] -4.151285e-07 -7.197897e-07 -4.239116e-06 -1.053320e-06  6.204834e-07
## [61,]  6.349848e-09  4.030100e-08 -1.060147e-06 -7.063822e-07 -1.655201e-06
## [62,]  1.422012e-07  8.264272e-07  7.241468e-07 -1.607804e-06 -1.515600e-05
## [63,]  4.687400e-07  8.490124e-07  7.041364e-07  1.453275e-07 -8.704487e-06
## [64,] -7.380616e-07  6.158478e-07  6.698950e-07  3.137329e-07 -1.619122e-06
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## [66,] -7.463512e-06  1.460566e-05  2.247994e-05  9.804547e-06 -9.934329e-06
##               [,51]         [,52]         [,53]         [,54]         [,55]
##  [1,]  2.071709e-07 -2.038103e-06  3.406853e-07  6.654122e-06  5.877754e-07
##  [2,]  2.228181e-07 -7.384519e-08  8.504126e-08 -2.006310e-06 -6.771396e-07
##  [3,]  1.949602e-07 -9.526350e-09  4.122186e-08  3.074057e-07 -2.749600e-06
##  [4,]  1.626013e-07 -7.110338e-08  4.218652e-08  8.614761e-07 -3.449398e-07
##  [5,]  2.738781e-07 -1.640892e-07  8.736454e-08  2.060453e-06  8.684285e-07
##  [6,]  1.871972e-07 -9.120650e-08  5.657706e-08  1.382405e-06  5.905910e-07
##  [7,]  1.352034e-07  2.135625e-08  1.691866e-08  5.707265e-07  3.371202e-07
##  [8,]  1.640622e-07 -1.126768e-07  4.593994e-08  9.098340e-07  4.635837e-07
##  [9,]  8.424377e-08 -1.189516e-07  4.080069e-08  6.159386e-07  2.966482e-07
## [10,] -3.738133e-07 -2.045166e-07  9.051308e-08  8.508736e-07  5.511678e-07
## [11,] -8.619167e-07 -3.840850e-07 -1.328912e-07  1.154484e-06  5.755211e-07
## [12,] -1.276665e-07 -5.534860e-07  4.022511e-08  1.665468e-08  4.311271e-07
## [13,] -1.785099e-07  1.192660e-08 -3.523298e-07  2.678160e-07 -3.118368e-08
## [14,] -2.035078e-06 -2.645357e-06  4.758640e-07  3.137169e-06 -2.834376e-07
## [15,] -2.655996e-06 -2.160539e-06  2.334192e-08 -4.029583e-07 -6.546455e-06
## [16,] -1.030496e-05 -1.119531e-05  2.762244e-06  5.044793e-05 -6.278792e-06
## [17,] -1.357657e-05 -9.018254e-07 -2.495408e-06 -8.426471e-05 -3.788615e-05
## [18,]  2.142936e-07 -6.593573e-07  4.053066e-07  4.826079e-06  1.823378e-07
## [19,]  2.802819e-07  1.162043e-08  4.095445e-08  1.025338e-06 -1.514062e-06
## [20,]  1.051331e-07  1.289639e-07 -2.842925e-08 -5.466062e-07 -6.219568e-07
## [21,] -6.675680e-08  2.463119e-07 -1.092586e-07 -2.596819e-06 -6.547574e-07
## [22,]  1.859184e-08  1.313031e-07 -5.740604e-08 -1.381563e-06 -2.594616e-07
## [23,]  1.496162e-07  1.250878e-07 -1.603083e-08 -6.245206e-08  2.164174e-07
## [24,]  8.613689e-08  2.255376e-07 -3.921166e-08 -3.324395e-07  5.192587e-08
## [25,] -6.942056e-09  1.152698e-07 -1.161996e-08 -1.895262e-07  5.233199e-09
## [26,] -9.430926e-07 -3.023790e-07  1.665966e-07  1.151382e-06  8.686536e-07
## [27,] -1.840999e-06 -4.929527e-07 -3.217512e-07  1.101964e-06  8.641482e-07
## [28,] -3.925516e-07 -1.086817e-06  1.955203e-07  7.521414e-07  5.590533e-07
## [29,] -3.272395e-07  1.901244e-07 -5.903691e-07 -1.573108e-07 -3.994475e-08
## [30,]  6.855576e-07  6.322135e-07 -1.602470e-07 -1.500402e-05 -3.334986e-06
## [31,]  7.562487e-07  3.215301e-07  3.162457e-08 -1.211704e-06 -1.108275e-05
## [32,]  4.055767e-07  3.998505e-08  7.584082e-08  1.978843e-06 -1.970608e-06
## [33,]  5.983249e-07  6.976673e-08  1.577699e-07  5.435712e-06  2.078081e-06
## [34,]  3.800437e-07  1.003800e-07  8.609202e-08  3.189851e-06  1.261379e-06
## [35,]  2.676613e-07  2.229780e-07  2.555938e-10  8.551447e-07  5.991060e-07
## [36,]  2.382349e-07  1.427327e-07  6.715513e-09  7.230974e-07  5.293449e-07
## [37,]  6.442480e-08  2.573549e-08  2.994976e-08  4.697526e-07  3.143642e-07
## [38,] -1.837701e-06 -5.782582e-07  3.386214e-07  2.425646e-06  1.773053e-06
## [39,] -3.495743e-06 -1.266774e-06 -4.090963e-07  3.603348e-06  1.808921e-06
## [40,] -9.532017e-07 -1.310331e-06  3.495864e-08 -1.548005e-06  1.009766e-06
## [41,] -5.260970e-07 -1.243038e-08 -7.357028e-07  1.501211e-06 -4.671475e-08
## [42,]  2.444701e-07 -3.559390e-06  1.947036e-06  4.042948e-05  6.814852e-07
## [43,]  1.278223e-06 -3.148425e-07  3.816061e-07  9.833220e-06 -1.042498e-05
## [44,]  3.585210e-07  3.971234e-07 -8.377871e-08 -1.927098e-06 -3.788115e-06
## [45,] -4.317852e-07  6.924829e-07 -3.969912e-07 -1.029014e-05 -2.829246e-06
## [46,] -6.341564e-08  3.656655e-07 -1.958629e-07 -5.038421e-06 -1.140386e-06
## [47,]  3.859161e-07  3.961479e-07 -4.908911e-08 -8.665338e-08  5.759170e-07
## [48,]  3.313809e-07  3.859590e-07 -4.169099e-08  1.723785e-07  5.400821e-07
## [49,]  5.492312e-08  1.530260e-07  2.392137e-08  2.698182e-07  3.608912e-07
## [50,] -3.956906e-06 -1.235237e-06  7.373083e-07  4.900000e-06  3.602323e-06
## [51,]  3.929425e-04 -2.160655e-06 -8.506514e-07  3.039225e-06  3.543906e-06
## [52,] -2.160655e-06  3.965856e-04  6.023115e-07  7.106430e-06  2.283535e-06
## [53,] -8.506514e-07  6.023115e-07  3.984928e-04 -3.029212e-06 -3.198640e-07
## [54,]  3.039225e-06  7.106430e-06 -3.029212e-06  2.863377e-04 -1.690354e-05
## [55,]  3.543906e-06  2.283535e-06 -3.198640e-07 -1.690354e-05  3.532619e-04
## [56,]  1.492912e-06  1.521377e-07  2.825644e-07  7.135233e-06 -8.826379e-06
## [57,]  2.061788e-06  2.643030e-07  6.088534e-07  2.116091e-05  7.677978e-06
## [58,]  1.229619e-06  4.097434e-07  2.939360e-07  1.131544e-05  4.266132e-06
## [59,]  7.585368e-07  6.831765e-07 -1.408938e-08  2.186861e-06  1.637977e-06
## [60,]  6.125318e-07  5.463727e-07 -2.453324e-08  1.300159e-06  1.225784e-06
## [61,]  1.233323e-07  1.960353e-07  7.429885e-08  1.006297e-06  8.329973e-07
## [62,] -8.283019e-06 -2.577130e-06  1.578490e-06  1.018218e-05  7.420535e-06
## [63,] -1.425352e-05 -5.677956e-06 -7.187341e-07  1.812084e-05  7.933165e-06
## [64,] -4.522794e-06 -3.390230e-06 -3.433997e-07 -1.389961e-05  3.057125e-06
## [65,] -1.453462e-06 -5.866457e-07 -1.736105e-06  1.039578e-05  1.547880e-07
## [66,] -1.595515e-05  3.625589e-05 -8.523298e-06 -1.145046e-04 -4.958553e-05
##               [,56]         [,57]         [,58]         [,59]         [,60]
##  [1,] -1.225256e-06 -5.018402e-06 -4.221252e-06 -2.988899e-06 -2.408519e-06
##  [2,]  7.376936e-08  4.355426e-07  2.368078e-07  7.226641e-08  5.283806e-08
##  [3,] -6.448106e-07  2.710203e-07  1.471119e-07  4.569579e-08  3.813737e-08
##  [4,] -9.709896e-07 -1.173946e-06 -8.083826e-08 -5.569023e-08 -3.149796e-08
##  [5,] -8.948254e-07 -3.798048e-06 -1.500448e-06 -1.748225e-07 -6.805998e-08
##  [6,]  1.715754e-07 -1.273372e-06 -1.992729e-06 -1.150191e-06 -1.513708e-07
##  [7,]  1.270496e-07  1.189811e-07 -1.002391e-06 -1.396940e-06 -3.140392e-07
##  [8,]  1.555114e-07 -1.687243e-07 -2.957081e-07 -4.667180e-07 -1.624907e-06
##  [9,]  9.432672e-08 -2.337995e-07 -2.035277e-07 -1.623801e-07 -5.619991e-07
## [10,]  1.873502e-07  2.206129e-08 -1.915303e-08  3.300283e-09  1.722302e-08
## [11,]  1.773287e-07 -1.756983e-07 -1.657271e-07 -7.666397e-08 -3.614804e-08
## [12,]  1.769495e-07  2.381651e-07  1.501298e-07  1.031346e-07  8.414379e-08
## [13,] -2.491334e-08 -1.051070e-07 -7.113805e-08 -3.468609e-08 -2.399877e-08
## [14,] -1.779394e-07  5.695182e-06  3.509972e-06 -7.035241e-07 -9.519382e-07
## [15,] -4.985554e-06 -8.866833e-06 -5.812330e-06 -3.224205e-06 -2.567351e-06
## [16,] -2.674165e-06  2.142850e-05  1.051319e-05 -3.900965e-06 -4.281018e-06
## [17,] -1.194133e-05 -2.470521e-05 -1.359908e-05 -4.631592e-06 -3.365966e-06
## [18,] -4.987145e-07 -8.684295e-07 -4.247013e-07  2.689912e-08  4.265312e-08
## [19,] -6.541308e-07 -5.301257e-07 -3.245104e-07 -1.628804e-08  2.008900e-08
## [20,] -5.997125e-08  4.309992e-07 -5.912401e-08  6.387016e-08  6.043807e-08
## [21,]  3.332958e-07  3.223392e-06  2.162261e-06  8.548765e-08  4.714146e-08
## [22,] -1.207692e-07  1.863611e-06  1.537983e-06 -5.458938e-07 -1.084068e-07
## [23,]  7.876975e-08  8.071581e-08 -4.691899e-07 -1.079924e-06 -2.511644e-07
## [24,]  3.409447e-08  3.376918e-07  1.719166e-07 -9.736516e-08 -1.161458e-06
## [25,]  1.057469e-08  2.228410e-07  1.839774e-07  1.036046e-07 -2.144264e-07
## [26,]  2.971640e-07  3.226996e-07  2.005198e-07  1.672678e-07  1.460785e-07
## [27,]  3.338230e-07  3.894950e-07  2.249367e-07  1.500677e-07  1.257839e-07
## [28,]  1.151858e-07  3.152214e-07  2.740641e-07  2.678429e-07  2.056810e-07
## [29,]  4.061419e-08  6.082253e-08  1.532146e-08 -2.796424e-08 -2.247028e-08
## [30,]  7.298997e-07  3.066483e-06  1.685436e-06  4.218915e-07  2.707661e-07
## [31,] -2.367687e-06  1.559633e-06  9.160042e-07  3.868689e-07  2.873282e-07
## [32,] -3.129932e-06 -3.155346e-06  3.299682e-07  1.737771e-07  1.256253e-07
## [33,] -3.194895e-06 -1.199785e-05 -4.120006e-06  3.340118e-07  3.290249e-07
## [34,]  3.502535e-07 -3.935477e-06 -5.316665e-06 -2.427022e-06 -6.664226e-08
## [35,]  2.421632e-07  6.372363e-07 -2.251115e-06 -3.276068e-06 -5.379706e-07
## [36,]  2.000907e-07  4.392825e-07 -1.939002e-08 -5.722497e-07 -3.324404e-06
## [37,]  1.119853e-07  1.426035e-07  8.983962e-08 -1.566573e-08 -9.826514e-07
## [38,]  6.127218e-07  7.787123e-07  4.882983e-07  3.652780e-07  3.055832e-07
## [39,]  5.357647e-07  5.183147e-07  3.265565e-07  3.066522e-07  2.612635e-07
## [40,]  5.772158e-07  1.350634e-06  8.865828e-07  5.035681e-07  3.725225e-07
## [41,] -1.380356e-07 -3.781089e-07 -2.287855e-07 -8.280607e-08 -5.568852e-08
## [42,] -3.924654e-06 -6.988689e-06 -3.636853e-06 -3.322107e-07 -9.998814e-08
## [43,] -3.970126e-06 -2.288495e-06 -1.444354e-06 -6.274099e-08  9.288262e-08
## [44,]  4.335252e-07  2.667117e-06 -2.027422e-07  1.835974e-07  1.832587e-07
## [45,]  2.150008e-06  1.511396e-05  8.684625e-06 -4.114046e-08 -3.928512e-08
## [46,] -4.520036e-07  8.285830e-06  5.937705e-06 -2.130388e-06 -4.151285e-07
## [47,]  2.185525e-07  6.829239e-07 -1.500440e-06 -3.470666e-06 -7.197897e-07
## [48,]  2.135529e-07  6.418158e-07  1.371251e-07 -5.903624e-07 -4.239116e-06
## [49,]  1.350443e-07  3.451654e-07  2.638460e-07  9.952836e-08 -1.053320e-06
## [50,]  1.238242e-06  1.537019e-06  9.636324e-07  7.356637e-07  6.204834e-07
## [51,]  1.492912e-06  2.061788e-06  1.229619e-06  7.585368e-07  6.125318e-07
## [52,]  1.521377e-07  2.643030e-07  4.097434e-07  6.831765e-07  5.463727e-07
## [53,]  2.825644e-07  6.088534e-07  2.939360e-07 -1.408938e-08 -2.453324e-08
## [54,]  7.135233e-06  2.116091e-05  1.131544e-05  2.186861e-06  1.300159e-06
## [55,] -8.826379e-06  7.677978e-06  4.266132e-06  1.637977e-06  1.225784e-06
## [56,]  3.892620e-04 -1.099169e-05  1.159621e-06  6.586580e-07  4.737886e-07
## [57,] -1.099169e-05  3.539400e-04 -1.663712e-05  1.628528e-06  1.351195e-06
## [58,]  1.159621e-06 -1.663712e-05  3.816944e-04 -6.327795e-06  7.027673e-08
## [59,]  6.586580e-07  1.628528e-06 -6.327795e-06  3.908538e-04 -1.380250e-06
## [60,]  4.737886e-07  1.351195e-06  7.027673e-08 -1.380250e-06  3.913876e-04
## [61,]  3.044276e-07  6.348103e-07  4.498407e-07  1.145545e-07 -2.405303e-06
## [62,]  2.551379e-06  3.221929e-06  2.016932e-06  1.514745e-06  1.268354e-06
## [63,]  2.024426e-06  1.945005e-06  1.319512e-06  1.391198e-06  1.175185e-06
## [64,]  2.802739e-06  5.971415e-06  3.650888e-06  1.689481e-06  1.231728e-06
## [65,] -9.392371e-07 -2.094561e-06 -1.163144e-06 -2.846442e-07 -1.805782e-07
## [66,] -1.236093e-05  9.697514e-05  7.742444e-05  4.605459e-05  3.035617e-05
##               [,61]         [,62]         [,63]         [,64]         [,65]
##  [1,] -9.087086e-07 -4.699933e-07 -3.936609e-07 -1.870912e-06 -2.623988e-07
##  [2,]  3.642180e-08  3.232340e-07  5.705443e-07 -4.238568e-07  3.282216e-07
##  [3,]  3.189334e-08  3.296436e-07  2.722229e-07  2.946487e-07 -1.039126e-07
##  [4,]  4.311014e-09  2.646690e-07  2.192270e-07  1.478002e-07 -8.065800e-08
##  [5,]  1.152071e-09  4.230983e-07  3.274238e-07  2.671780e-07 -1.634222e-07
##  [6,]  1.002863e-08  2.939104e-07  2.271826e-07  2.132891e-07 -1.137247e-07
##  [7,] -4.275814e-09  2.455000e-07  2.191335e-07  1.846603e-07 -5.225543e-08
##  [8,] -4.882826e-07  2.660183e-07  2.300484e-07  5.536494e-08 -6.269617e-08
##  [9,] -3.025212e-07 -1.575100e-07  9.387797e-08 -3.538134e-08 -2.644769e-08
## [10,] -2.901002e-07 -1.871614e-06 -9.293262e-07 -2.251565e-07  7.304006e-08
## [11,] -3.438891e-08 -9.055465e-07 -1.835181e-06 -4.835596e-07 -4.107966e-07
## [12,]  3.893198e-08 -6.087623e-08 -2.615107e-07 -1.002105e-06  1.344995e-07
## [13,]  1.232655e-09  1.263225e-07 -3.499150e-07  1.484781e-07 -5.750123e-07
## [14,] -5.381691e-07 -4.557152e-06 -4.848405e-06 -2.461889e-06 -3.900716e-07
## [15,] -1.279244e-06 -5.646722e-06 -5.345635e-06 -4.270122e-06  3.011435e-07
## [16,] -2.816439e-06 -2.404866e-05 -2.820111e-05 -4.957648e-06 -4.255576e-06
## [17,] -2.423329e-06 -2.317753e-05 -1.903181e-05 -2.326876e-05  7.821308e-06
## [18,]  7.060323e-09 -1.407543e-07 -2.593597e-07  3.029971e-07 -1.505735e-07
## [19,]  3.427589e-08  4.934026e-07  4.421804e-07  3.388137e-07 -1.051859e-07
## [20,]  3.434482e-08  2.454825e-07  2.795615e-07  9.307792e-08  3.258090e-08
## [21,]  1.956972e-08  1.759972e-08  1.741887e-07 -2.830533e-07  2.265740e-07
## [22,]  1.261988e-08  1.144554e-07  2.064756e-07 -1.538591e-07  1.261473e-07
## [23,]  1.396718e-08  3.163913e-07  3.302886e-07  1.830531e-07 -1.008803e-09
## [24,] -2.387686e-07  2.278124e-07  2.360548e-07  2.777176e-07  3.991976e-09
## [25,] -1.274245e-07 -4.364556e-07  3.536161e-08  1.376160e-07  2.015816e-08
## [26,] -4.547424e-07 -3.639177e-06 -1.908531e-06 -3.526316e-07  1.772738e-07
## [27,]  6.243348e-09 -1.855682e-06 -3.519433e-06 -1.014600e-06 -5.789598e-07
## [28,]  7.689765e-08 -4.105976e-07 -1.079069e-06 -1.219440e-06 -3.266018e-08
## [29,]  1.337061e-08  3.241082e-07 -4.183381e-07  1.316605e-08 -7.313694e-07
## [30,]  1.797335e-07  1.572402e-06  2.802624e-06 -2.063006e-06  1.615590e-06
## [31,]  1.792454e-07  1.446178e-06  1.372505e-06  1.076587e-06 -2.148892e-07
## [32,]  8.187929e-08  6.951382e-07  5.562663e-07  7.442066e-07 -2.482360e-07
## [33,]  1.659562e-07  9.622635e-07  6.438341e-07  1.534439e-06 -5.315536e-07
## [34,]  1.246521e-07  6.313910e-07  4.402311e-07  1.004665e-06 -3.223687e-07
## [35,]  3.320255e-08  5.283783e-07  4.795833e-07  5.894727e-07 -1.062814e-07
## [36,] -9.545774e-07  4.708211e-07  4.264268e-07  4.236327e-07 -8.137951e-08
## [37,] -5.713729e-07 -8.099735e-07  1.020723e-07  1.615463e-07 -8.717895e-09
## [38,] -8.970551e-07 -7.532276e-06 -4.140365e-06 -7.536374e-07  3.991930e-07
## [39,]  1.930788e-08 -4.122888e-06 -7.425711e-06 -1.808145e-06 -1.171072e-06
## [40,]  1.550770e-07 -7.109563e-07 -1.736764e-06 -2.889926e-06  3.912976e-07
## [41,]  1.122687e-08  5.097315e-07 -1.052652e-06  4.232365e-07 -1.455154e-06
## [42,] -1.553834e-07 -2.219269e-06 -4.324336e-06  4.208172e-06 -2.775708e-06
## [43,]  1.427460e-07  1.983805e-06  1.346999e-06  2.411112e-06 -1.030571e-06
## [44,]  1.083718e-07  8.186536e-07  9.167408e-07  3.258134e-07  8.561463e-08
## [45,] -1.967593e-08 -3.068923e-07  3.165582e-07 -1.576663e-06  9.299378e-07
## [46,]  6.349848e-09  1.422012e-07  4.687400e-07 -7.380616e-07  4.661840e-07
## [47,]  4.030100e-08  8.264272e-07  8.490124e-07  6.158478e-07 -2.451638e-08
## [48,] -1.060147e-06  7.241468e-07  7.041364e-07  6.698950e-07 -5.607772e-08
## [49,] -7.063822e-07 -1.607804e-06  1.453275e-07  3.137329e-07  2.721442e-08
## [50,] -1.655201e-06 -1.515600e-05 -8.704487e-06 -1.619122e-06  8.887222e-07
## [51,]  1.233323e-07 -8.283019e-06 -1.425352e-05 -4.522794e-06 -1.453462e-06
## [52,]  1.960353e-07 -2.577130e-06 -5.677956e-06 -3.390230e-06 -5.866457e-07
## [53,]  7.429885e-08  1.578490e-06 -7.187341e-07 -3.433997e-07 -1.736105e-06
## [54,]  1.006297e-06  1.018218e-05  1.812084e-05 -1.389961e-05  1.039578e-05
## [55,]  8.329973e-07  7.420535e-06  7.933165e-06  3.057125e-06  1.547880e-07
## [56,]  3.044276e-07  2.551379e-06  2.024426e-06  2.802739e-06 -9.392371e-07
## [57,]  6.348103e-07  3.221929e-06  1.945005e-06  5.971415e-06 -2.094561e-06
## [58,]  4.498407e-07  2.016932e-06  1.319512e-06  3.650888e-06 -1.163144e-06
## [59,]  1.145545e-07  1.514745e-06  1.391198e-06  1.689481e-06 -2.846442e-07
## [60,] -2.405303e-06  1.268354e-06  1.175185e-06  1.231728e-06 -1.805782e-07
## [61,]  3.983826e-04 -3.121018e-06  2.429420e-07  5.710472e-07  1.983967e-08
## [62,] -3.121018e-06  3.689960e-04 -1.860859e-05 -3.505450e-06  1.982417e-06
## [63,]  2.429420e-07 -1.860859e-05  3.689416e-04 -7.585466e-06 -3.461285e-06
## [64,]  5.710472e-07 -3.505450e-06 -7.585466e-06  3.897564e-04  1.666173e-06
## [65,]  1.983967e-08  1.982417e-06 -3.461285e-06  1.666173e-06  3.954000e-04
## [66,]  9.249532e-06 -1.997825e-05 -1.765605e-05  3.000705e-05  4.186419e-06
##               [,66]
##  [1,] -1.260574e-04
##  [2,] -3.626361e-06
##  [3,] -6.311082e-06
##  [4,] -1.493025e-05
##  [5,] -2.623247e-05
##  [6,] -1.526203e-05
##  [7,] -4.300049e-06
##  [8,] -2.218876e-05
##  [9,] -1.939114e-05
## [10,] -1.241618e-05
## [11,] -1.998392e-05
## [12,] -1.397648e-06
## [13,] -2.320375e-06
## [14,]  5.701084e-05
## [15,]  8.019525e-05
## [16,]  9.052213e-05
## [17,]  1.570192e-04
## [18,]  1.017970e-05
## [19,] -1.238721e-05
## [20,]  1.297193e-06
## [21,]  7.913723e-06
## [22,] -1.554536e-06
## [23,]  9.969276e-07
## [24,]  1.880251e-05
## [25,]  1.202901e-05
## [26,] -3.574090e-06
## [27,] -6.212121e-06
## [28,]  1.501326e-05
## [29,] -3.646391e-06
## [30,] -8.109113e-06
## [31,] -3.141515e-06
## [32,] -4.360917e-06
## [33,]  1.755875e-05
## [34,]  1.581941e-05
## [35,]  1.449813e-05
## [36,]  4.450066e-06
## [37,] -2.015044e-06
## [38,] -4.769992e-06
## [39,] -8.082144e-06
## [40,]  1.737034e-05
## [41,] -3.047119e-06
## [42,]  5.023227e-05
## [43,] -4.959046e-05
## [44,]  2.045903e-06
## [45,]  9.203763e-06
## [46,] -7.463512e-06
## [47,]  1.460566e-05
## [48,]  2.247994e-05
## [49,]  9.804547e-06
## [50,] -9.934329e-06
## [51,] -1.595515e-05
## [52,]  3.625589e-05
## [53,] -8.523298e-06
## [54,] -1.145046e-04
## [55,] -4.958553e-05
## [56,] -1.236093e-05
## [57,]  9.697514e-05
## [58,]  7.742444e-05
## [59,]  4.605459e-05
## [60,]  3.035617e-05
## [61,]  9.249532e-06
## [62,] -1.997825e-05
## [63,] -1.765605e-05
## [64,]  3.000705e-05
## [65,]  4.186419e-06
## [66,]  5.400261e-03
## 
## $log_evidence
## [1] -131.93
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 134

3i)

Use the viz_post_coefs() function to visualize the posterior coefficient summaries for model 3 and model 6, based on the very strong prior specification.

SOLUTION

post_means_3_2 <- laplace_03_very_strong$mode[1:ncol(X03)]
post_sds_3_2 <- sqrt(diag(laplace_03_very_strong$var_matrix)[1:ncol(X03)])
viz_post_coefs(post_means_3_2, post_sds_3_2, xnames_3)

post_means_6_2 <- laplace_06_very_strong$mode[1:ncol(X06)]
post_sds_6_2 <- sqrt(diag(laplace_06_very_strong$var_matrix)[1:ncol(X06)])
viz_post_coefs(post_means_6_2, post_sds_6_2, xnames_6)

3j)

Describe the influence of the regression coefficient prior standard deviation on the coefficient posterior distributions.

SOLUTION

What do you think?
The prior standard deviation of regression coefficients affects how much the prior information influences the coefficient posterior distributions. A diffuse or weak prior with a larger standard deviation provides a wider range of possible values for the coefficients, whereas a strong or informative prior with a smaller standard deviation constrains the range of possible values, giving more weight to the prior information. This leads to more narrowly distributed and less variable coefficient posterior distributions with a strong prior, indicating a lower chance of overfitting to the data. Hence, the prior standard deviation plays a crucial role in balancing the influence of the prior knowledge and the data in the model fitting process.

3k)

You previously compared the two models using the Bayes Factor based on the weak prior specification.

Compare the performance of the two models with Bayes Factors again, but considering the results based on the strong and very strong priors. Does the prior influence which model is considered to be better?

SOLUTIOn

bf1 <- exp(laplace_03_strong$log_evidence - laplace_06_strong$log_evidence)
bf1

## [1] 2.672262e+13

bf1 is greater than 1, then Model 3 is favored over Model 6.

bf2 <- exp(laplace_03_very_strong$log_evidence - laplace_06_very_strong$log_evidence)
bf2

## [1] 0.0002438238

bf2 is smaller than 1, then Model 6 is favored over Model 3.

Problem 04

You examined the behavior of the coefficient posterior based on the influence of the prior. Let’s now consider the prior’s influence by examining the posterior predictive distributions.

4a)

You will make posterior predictions following the approach from the previous assignment. Posterior samples are generated and those samples are used to calculate the posterior samples of the mean trend and generate random posterior samples of the response around the mean. In the previous assignment, you made posterior predictions in order to calculate errors. In this assignment, you will not calculate errors, instead you will summarize the posterior predictions of the mean and of the random response.

The generate_lm_post_samples() function is defined for you below. It uses the MASS::mvrnorm() function generate posterior samples from the Laplace Approximation’s MVN distribution.

generate_lm_post_samples <- function(mvn_result, length_beta, num_samples)
{
  MASS::mvrnorm(n = num_samples,
                mu = mvn_result$mode,
                Sigma = mvn_result$var_matrix) %>% 
    as.data.frame() %>% tibble::as_tibble() %>% 
    purrr::set_names(c(sprintf("beta_%02d", 0:(length_beta-1)), "varphi")) %>% 
    mutate(sigma = exp(varphi))
}

The code chunk below starts the post_lm_pred_samples() function. This function generates posterior mean trend predictions and posterior predictions of the response. The first argument, Xnew, is a potentially new or test design matrix that we wish to make predictions at. The second argument, Bmat, is a matrix of posterior samples of the \(\boldsymbol{\beta}\)-parameters, and the third argument, sigma_vector, is a vector of posterior samples of the likelihood noise. The Xnew matrix has rows equal to the number of predictions points, M, and the Bmat matrix has rows equal to the number of posterior samples S.

You must complete the function by performing the necessary matrix math to calculate the matrix of posterior mean trend predictions, Umat, and the matrix of posterior response predictions, Ymat. You must also complete missing arguments to the definition of the Rmat and Zmat matrices. The Rmat matrix replicates the posterior likelihood noise samples the correct number of times. The Zmat matrix is the matrix of randomly generated standard normal values. You must correctly specify the required number of rows to the Rmat and Zmat matrices.

The post_lm_pred_samples() returns the Umat and Ymat matrices contained within a list.

Perform the necessary matrix math to calculate the matrix of posterior predicted mean trends Umat and posterior predicted responses Ymat. You must specify the number of required rows to create the Rmat and Zmat matrices.

HINT: The following code chunk should look famaliar…

SOLUTION

post_lm_pred_samples <- function(Xnew, Bmat, sigma_vector)
{
  # number of new prediction locations
  M <- nrow(Xnew)
  # number of posterior samples
  S <- nrow(Bmat)
  
  # matrix of linear predictors
  Umat <- Xnew %*% t(Bmat)
  
  # assmeble matrix of sigma samples, set the number of rows
  Rmat <- matrix(rep(sigma_vector, M), M, byrow = TRUE)
  
  # generate standard normal and assemble into matrix
  # set the number of rows
  Zmat <- matrix(rnorm(M*S), M, byrow = TRUE)
  
  # calculate the random observation predictions
  Ymat <- Umat + Rmat * Zmat
  
  # package together
  list(Umat = Umat, Ymat = Ymat)
}

4b)

Since this assignment is focused on visualizing the predictions, we will summarize the posterior predictions to focus on the posterior means and the middle 95% uncertainty intervals. The code chunk below is defined for you which serves as a useful wrapper function to call post_lm_pred_samples().

make_post_lm_pred <- function(Xnew, post)
{
  Bmat <- post %>% select(starts_with("beta_")) %>% as.matrix()
  
  sigma_vector <- post %>% pull(sigma)
  
  post_lm_pred_samples(Xnew, Bmat, sigma_vector)
}

The code chunk below defines a function summarize_lm_pred_from_laplace() which manages the actions necessary to summarize posterior predictions. The first argument, mvn_result, is the Laplace Approximation object. The second object is the test design matrix, Xtest, and the third argument, num_samples, is the number of posterior samples to make.

You must complete the code chunk below which summarizes the posterior predictions. This function takes care of most of the coding for you. You do not have to worry about the generation of the posterior samples OR calculating the posterior quantiles associated with the middle 95% uncertainty interval. You must calculate the posterior average by deciding on whether you should use colMeans() or rowMeans() to calculate the average across all posterior samples per prediction location.

Follow the comments in the code chunk below to complete the definition of the summarize_lm_pred_from_laplace() function. You must calculate the average posterior mean trend and the average posterior response.

SOLUTION

summarize_lm_pred_from_laplace <- function(mvn_result, Xtest, num_samples)
{
  # generate posterior samples of the beta parameters
  post <- generate_lm_post_samples(mvn_result, ncol(Xtest), num_samples)
  
  # make posterior predictions on the test set
  pred_test <- make_post_lm_pred(Xtest, post)
  
  # calculate summary statistics on the predicted mean and response
  # summarize over the posterior samples
  
  # posterior mean, should you summarize along rows (rowMeans) or 
  # summarize down columns (colMeans) ???
  mu_avg <- rowMeans(pred_test$Umat)
  y_avg <- rowMeans(pred_test$Umat)
  
  # posterior quantiles for the middle 95% uncertainty intervals
  mu_lwr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.025)
  mu_upr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.975)
  y_lwr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.025)
  y_upr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.975)
  
  # book keeping
  tibble::tibble(
    mu_avg = mu_avg,
    mu_lwr = mu_lwr,
    mu_upr = mu_upr,
    y_avg = y_avg,
    y_lwr = y_lwr,
    y_upr = y_upr
  ) %>% 
    tibble::rowid_to_column("pred_id")
}

4c)

When you made predictions in Problem 02, the lm() object handled making the test design matrix. However, since we have programmed the Bayesian modeling approach from scratch we need to create the test design matrix manually.

Create the test design matrix based on the visualization grid, viz_grid, using the model 3 formulation. Assign the result to the X03_test object.

Call the summarize_lm_pred_from_laplace() function to summarize the posterior predictions from the model 3 formulation for the weak, strong, and very strong prior specifications. Use 5000 posterior samples for each case. Assign the results from the weak prior to post_pred_summary_viz_03_weak, the results from the strong prior to post_pred_summary_viz_03_strong, and the results from the very strong prior to post_pred_summary_viz_03_very_strong.

SOLUTION

X03_test <- model.matrix( ~ (x1 + I(x1^2)) * (x2 + I(x2^2)), viz_grid)

post_pred_summary_viz_03_weak <- summarize_lm_pred_from_laplace(laplace_03_weak,X03_test, 5000)
post_pred_summary_viz_03_weak

## # A tibble: 909 × 7
##    pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr
##      <int>  <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl>
##  1       1  -1.13  -9.60   7.47 -1.13 -9.59  7.55
##  2       2  -1.19  -9.26   7.07 -1.19 -9.37  7.17
##  3       3  -1.25  -8.93   6.65 -1.25 -8.96  6.76
##  4       4  -1.31  -8.64   6.21 -1.31 -8.75  6.24
##  5       5  -1.37  -8.36   5.75 -1.37 -8.52  5.90
##  6       6  -1.44  -8.08   5.31 -1.44 -8.19  5.50
##  7       7  -1.50  -7.78   4.90 -1.50 -8.02  5.15
##  8       8  -1.56  -7.51   4.51 -1.56 -7.68  4.61
##  9       9  -1.62  -7.28   4.16 -1.62 -7.64  4.27
## 10      10  -1.68  -7.07   3.79 -1.68 -7.25  3.93
## # … with 899 more rows

post_pred_summary_viz_03_strong <- summarize_lm_pred_from_laplace(laplace_03_strong,X03_test, 5000)
post_pred_summary_viz_03_strong 

## # A tibble: 909 × 7
##    pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr
##      <int>  <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl>
##  1       1  -1.21  -9.59   6.85 -1.21 -9.81  6.97
##  2       2  -1.27  -9.30   6.44 -1.27 -9.26  6.50
##  3       3  -1.33  -9.00   6.03 -1.33 -9.15  6.18
##  4       4  -1.39  -8.70   5.63 -1.39 -8.93  5.83
##  5       5  -1.45  -8.40   5.24 -1.45 -8.43  5.37
##  6       6  -1.51  -8.16   4.87 -1.51 -8.20  4.97
##  7       7  -1.57  -7.87   4.49 -1.57 -8.03  4.68
##  8       8  -1.63  -7.63   4.11 -1.63 -7.73  4.24
##  9       9  -1.69  -7.35   3.75 -1.69 -7.55  4.00
## 10      10  -1.75  -7.13   3.39 -1.75 -7.31  3.60
## # … with 899 more rows

post_pred_summary_viz_03_very_strong <- summarize_lm_pred_from_laplace(laplace_03_very_strong,X03_test, 5000)
post_pred_summary_viz_03_very_strong 

## # A tibble: 909 × 7
##    pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr
##      <int>  <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl>
##  1       1  -3.59  -7.07 -0.103 -3.59 -7.52 0.431
##  2       2  -3.47  -6.82 -0.110 -3.47 -7.29 0.441
##  3       3  -3.34  -6.56 -0.120 -3.34 -6.94 0.511
##  4       4  -3.22  -6.31 -0.121 -3.22 -6.75 0.311
##  5       5  -3.10  -6.06 -0.119 -3.10 -6.70 0.338
##  6       6  -2.99  -5.83 -0.125 -2.99 -6.24 0.319
##  7       7  -2.88  -5.59 -0.124 -2.88 -6.14 0.387
##  8       8  -2.76  -5.37 -0.127 -2.76 -5.96 0.402
##  9       9  -2.66  -5.15 -0.139 -2.66 -5.68 0.495
## 10      10  -2.55  -4.93 -0.144 -2.55 -5.42 0.436
## # … with 899 more rows

4d)

You will now visualize the posterior predictions from the model 3 Bayesian models associated with the weak, strong, and very strong priors. The viz_grid object is joined to the prediction dataframes assuming you have used the correct variable names!

Visualize the predicted means, confidence intervals, and prediction intervals in the style of those that you created in Problem 02. The confidence interval bounds are mu_lwr and mu_upr columns and the prediction interval bounds are the y_lwr and y_upr columns, respectively. The posterior predicted mean of the mean is mu_avg.

Pipe the result of the joined dataframe into ggplot() and make appropriate aesthetics and layers to visualize the predictions with the x1 variable mapped to the x aesthetic and the x2 variable used as a facet variable.

SOLUTION

post_pred_summary_viz_03_weak %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')

## # A tibble: 909 × 9
##    pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr    x1    x2
##      <int>  <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
##  1       1  -1.13  -9.60   7.47 -1.13 -9.59  7.55 -3.2     -3
##  2       2  -1.19  -9.26   7.07 -1.19 -9.37  7.17 -3.14    -3
##  3       3  -1.25  -8.93   6.65 -1.25 -8.96  6.76 -3.07    -3
##  4       4  -1.31  -8.64   6.21 -1.31 -8.75  6.24 -3.01    -3
##  5       5  -1.37  -8.36   5.75 -1.37 -8.52  5.90 -2.94    -3
##  6       6  -1.44  -8.08   5.31 -1.44 -8.19  5.50 -2.88    -3
##  7       7  -1.50  -7.78   4.90 -1.50 -8.02  5.15 -2.82    -3
##  8       8  -1.56  -7.51   4.51 -1.56 -7.68  4.61 -2.75    -3
##  9       9  -1.62  -7.28   4.16 -1.62 -7.64  4.27 -2.69    -3
## 10      10  -1.68  -7.07   3.79 -1.68 -7.25  3.93 -2.62    -3
## # … with 899 more rows

post_pred_summary_viz_03_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')

## # A tibble: 909 × 9
##    pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr    x1    x2
##      <int>  <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
##  1       1  -1.21  -9.59   6.85 -1.21 -9.81  6.97 -3.2     -3
##  2       2  -1.27  -9.30   6.44 -1.27 -9.26  6.50 -3.14    -3
##  3       3  -1.33  -9.00   6.03 -1.33 -9.15  6.18 -3.07    -3
##  4       4  -1.39  -8.70   5.63 -1.39 -8.93  5.83 -3.01    -3
##  5       5  -1.45  -8.40   5.24 -1.45 -8.43  5.37 -2.94    -3
##  6       6  -1.51  -8.16   4.87 -1.51 -8.20  4.97 -2.88    -3
##  7       7  -1.57  -7.87   4.49 -1.57 -8.03  4.68 -2.82    -3
##  8       8  -1.63  -7.63   4.11 -1.63 -7.73  4.24 -2.75    -3
##  9       9  -1.69  -7.35   3.75 -1.69 -7.55  4.00 -2.69    -3
## 10      10  -1.75  -7.13   3.39 -1.75 -7.31  3.60 -2.62    -3
## # … with 899 more rows

post_pred_summary_viz_03_very_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')

## # A tibble: 909 × 9
##    pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr    x1    x2
##      <int>  <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
##  1       1  -3.59  -7.07 -0.103 -3.59 -7.52 0.431 -3.2     -3
##  2       2  -3.47  -6.82 -0.110 -3.47 -7.29 0.441 -3.14    -3
##  3       3  -3.34  -6.56 -0.120 -3.34 -6.94 0.511 -3.07    -3
##  4       4  -3.22  -6.31 -0.121 -3.22 -6.75 0.311 -3.01    -3
##  5       5  -3.10  -6.06 -0.119 -3.10 -6.70 0.338 -2.94    -3
##  6       6  -2.99  -5.83 -0.125 -2.99 -6.24 0.319 -2.88    -3
##  7       7  -2.88  -5.59 -0.124 -2.88 -6.14 0.387 -2.82    -3
##  8       8  -2.76  -5.37 -0.127 -2.76 -5.96 0.402 -2.75    -3
##  9       9  -2.66  -5.15 -0.139 -2.66 -5.68 0.495 -2.69    -3
## 10      10  -2.55  -4.93 -0.144 -2.55 -5.42 0.436 -2.62    -3
## # … with 899 more rows

post_pred_summary_viz_03_weak %>% 
  right_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id') %>% 
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr,
                            ymax = y_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr,
                            ymax = mu_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) +
  coord_cartesian(ylim = c(-7, 7)) +
  facet_wrap(~x2)

post_pred_summary_viz_03_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')%>% 
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr,
                            ymax = y_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr,
                            ymax = mu_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) +
  coord_cartesian(ylim = c(-7, 7)) +
  facet_wrap(~x2)

post_pred_summary_viz_03_very_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')%>% 
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr,
                            ymax = y_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr,
                            ymax = mu_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) +
  coord_cartesian(ylim = c(-7, 7)) +
  facet_wrap(~x2)

### 4e)

In order to make posterior predictions for the model 6 formulation you must create a test design matrix consistent with the training set basis. The code chunk below creates a helper function which extracts the interior and boundary knots of a natural spline associated with the training set for you. The first argument, J, is the degrees-of-freedom (DOF) of the spline, the second argument, train_data, is the training data set. The third argument xname is the name of the variable you are applying the spline to. The xname argument must be provided as a character string.

make_splines_training_knots <- function(J, train_data, xname)
{
  # extract the input from the training set
  x <- train_data %>% select(all_of(xname)) %>% pull()
  
  # create the training basis
  train_basis <- splines::ns(x, df = J)
  
  # extract the knots
  interior_knots <- as.vector(attributes(train_basis)$knots)
  
  boundary_knots <- as.vector(attributes(train_basis)$Boundary.knots)
  
  # book keeping
  list(interior_knots = interior_knots,
       boundary_knots = boundary_knots)
}

Create the test design matrix based on the visualization grid, viz_grid, using the model 6 formulation. Assign the result to the X06_test object. Use the make_splines_training_knots() function to get the interior and boundary knots associated with the training set for the x1 variable to create the test design matrix.

Call the summarize_lm_pred_from_laplace() function to summarize the posterior predictions from the model 6 formulation for the weak, strong, and very strong prior specifications. Use 5000 posterior samples for each case. Assign the results from the weak prior to post_pred_summary_viz_06_weak, the results from the strong prior to post_pred_summary_viz_06_strong, and the results from the very strong prior to post_pred_summary_viz_06_very_strong.

HINT: The make_spline_training_knots() function returns a list! The fields or elements of the list can be accessed via the $ operator.

SOLUTION

# create test design matrix based on visualization grid for model 6
x1_knots <- make_splines_training_knots( 12, df, "x1")
X06_test <- model.matrix(~ splines::ns(x1, knots = x1_knots$interior_knots, Boundary.knots = x1_knots$boundary_knots) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)),  viz_grid)

# summarize posterior predictions for weak, strong, and very strong priors
post_pred_summary_viz_06_weak <- summarize_lm_pred_from_laplace(laplace_06_weak, X06_test, 5000)
post_pred_summary_viz_06_strong <- summarize_lm_pred_from_laplace(laplace_06_strong, X06_test,  5000)
post_pred_summary_viz_06_very_strong <- summarize_lm_pred_from_laplace(laplace_06_very_strong, X06_test, 5000)

4f)

You will now visualize the posterior predictions from the model 6 Bayesian models associated with the weak, strong, and very strong priors. The viz_grid object is joined to the prediction dataframes assuming you have used the correct variable names!

Visualize the predicted means, confidence intervals, and prediction intervals in the style of those that you created in Problem 02. The confidence interval bounds are mu_lwr and mu_upr columns and the prediction interval bounds are the y_lwr and y_upr columns, respectively. The posterior predicted mean of the mean is mu_avg.

Pipe the result of the joined dataframe into ggplot() and make appropriate aesthetics and layers to visualize the predictions with the x1 variable mapped to the x aesthetic and the x2 variable used as a facet variable.

SOLUTION

post_pred_summary_viz_06_weak %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')

## # A tibble: 909 × 9
##    pred_id mu_avg mu_lwr mu_upr y_avg  y_lwr y_upr    x1    x2
##      <int>  <dbl>  <dbl>  <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl>
##  1       1  -446. -1977.  1050. -446. -1977. 1050. -3.2     -3
##  2       2  -422. -1868.   989. -422. -1868.  988. -3.14    -3
##  3       3  -398. -1762.   930. -398. -1762.  929. -3.07    -3
##  4       4  -373. -1653.   872. -373. -1653.  871. -3.01    -3
##  5       5  -349. -1546.   815. -349. -1546.  815. -2.94    -3
##  6       6  -325. -1439.   753. -325. -1438.  753. -2.88    -3
##  7       7  -301. -1332.   699. -301. -1333.  699. -2.82    -3
##  8       8  -277. -1224.   647. -277. -1225.  647. -2.75    -3
##  9       9  -252. -1116.   587. -252. -1115.  587. -2.69    -3
## 10      10  -228. -1008.   531. -228. -1007.  531. -2.62    -3
## # … with 899 more rows

post_pred_summary_viz_06_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')

## # A tibble: 909 × 9
##    pred_id mu_avg mu_lwr mu_upr y_avg  y_lwr y_upr    x1    x2
##      <int>  <dbl>  <dbl>  <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl>
##  1       1  -36.3 -125.    53.5 -36.3 -126.   53.8 -3.2     -3
##  2       2  -34.7 -118.    49.8 -34.7 -118.   49.7 -3.14    -3
##  3       3  -33.1 -111.    45.4 -33.1 -111.   45.6 -3.07    -3
##  4       4  -31.5 -104.    41.8 -31.5 -104.   42.0 -3.01    -3
##  5       5  -29.9  -96.8   38.3 -29.9  -97.2  38.1 -2.94    -3
##  6       6  -28.3  -90.2   34.3 -28.3  -90.0  33.7 -2.88    -3
##  7       7  -26.7  -83.3   30.2 -26.7  -83.1  30.2 -2.82    -3
##  8       8  -25.1  -77.0   26.1 -25.1  -76.9  26.4 -2.75    -3
##  9       9  -23.5  -70.3   22.7 -23.5  -70.4  22.6 -2.69    -3
## 10      10  -21.8  -64.0   19.4 -21.8  -63.9  19.5 -2.62    -3
## # … with 899 more rows

post_pred_summary_viz_06_very_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')

## # A tibble: 909 × 9
##    pred_id mu_avg mu_lwr  mu_upr y_avg y_lwr   y_upr    x1    x2
##      <int>  <dbl>  <dbl>   <dbl> <dbl> <dbl>   <dbl> <dbl> <dbl>
##  1       1  -3.66  -7.22 -0.0909 -3.66 -7.60  0.312  -3.2     -3
##  2       2  -3.68  -7.04 -0.312  -3.68 -7.47  0.0557 -3.14    -3
##  3       3  -3.70  -6.88 -0.535  -3.70 -7.28 -0.124  -3.07    -3
##  4       4  -3.72  -6.75 -0.750  -3.72 -7.29 -0.397  -3.01    -3
##  5       5  -3.74  -6.60 -0.923  -3.74 -6.98 -0.470  -2.94    -3
##  6       6  -3.76  -6.46 -1.09   -3.76 -6.92 -0.613  -2.88    -3
##  7       7  -3.78  -6.30 -1.26   -3.78 -6.79 -0.823  -2.82    -3
##  8       8  -3.80  -6.18 -1.42   -3.80 -6.65 -1.03   -2.75    -3
##  9       9  -3.82  -6.09 -1.58   -3.82 -6.53 -1.02   -2.69    -3
## 10      10  -3.84  -6.03 -1.70   -3.84 -6.48 -1.15   -2.62    -3
## # … with 899 more rows

post_pred_summary_viz_06_weak %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')%>% 
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr,
                            ymax = y_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr,
                            ymax = mu_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) +
  coord_cartesian(ylim = c(-7, 7)) +
  facet_wrap(~x2)

post_pred_summary_viz_06_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')%>% 
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr,
                            ymax = y_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr,
                            ymax = mu_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) +
  coord_cartesian(ylim = c(-7, 7)) +
  facet_wrap(~x2)

post_pred_summary_viz_06_very_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id')%>% 
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr,
                            ymax = y_upr),
              fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr,
                            ymax = mu_upr),
              fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) +
  coord_cartesian(ylim = c(-7, 7)) +
  facet_wrap(~x2)

4g)

Describe the behavior of the predictions as the prior standard deviation decreased. Are the posterior predictions consistent with the behavior of the posterior coefficients?

SOLUTION

What do you think?
As the prior standard deviation gets smaller, we become more certain about the posterior distribution of the coefficients. This increased certainty in the coefficients results in more concentrated posterior predictions around the observed values.

The reason for this is that a smaller prior standard deviation places more emphasis on the observed data, which leads to more confidence in the estimates of the coefficients. This, in turn, leads to more confidence in the predictions based on these coefficients.

Overall, the behavior of the posterior predictions is consistent with the behavior of the posterior coefficients. As we become more certain about the coefficients, we also become more certain about the predictions that are based on these coefficients.

Problem 05

Now that you have worked with Bayesian models with the prior regularizing the coefficients, you will consider non-Bayesian regularization methods. You will work with the glmnet package in this problem which takes care of all fitting and visualization for you.

The code chunk below loads in glmnet and so you must have glmnet installed before running this code chunk. IMPORANT: the eval flag is set to FALSE below. Once you download glmnet set eval=TRUE.

library(glmnet)

## Loading required package: Matrix

## 
## Attaching package: 'Matrix'

## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack

## Loaded glmnet 4.1-7

5a)

glmnet does not work with the formula interface. And so you must create the training design matrix. However, glmnet prefers the the intercept column of ones to not be included in the design matrix. To support that you must define new design matrices. These matrices will use the same formulation but you must remove the intercept column. This is easy to do with the formula interface and the model.matrix() function. Include - 1 in the formula and model.matrix() will not include the intercept. The code chunk below demonstrates removing the intercept column for a model with linear additive features.

model.matrix( y ~ x1 + x2 - 1, data = df) %>% head()

##           x1         x2
## 1 -0.3092328  0.3087799
## 2  0.6312721 -0.5479198
## 3 -0.6827669  2.1664494
## 4  0.2693056  1.2097037
## 5  0.3725202  0.7854860
## 6  1.2966439 -0.1877231

Create the design matrices for glmnet for the model 3 and model 6 formulations. Remove the intercept column for both and assign the results to X03_glmnet and X06_glmnet.

SOLUTION

X03_glmnet <- model.matrix(y ~ (x1 + I(x1^2)) * (x2 + I(x2^2)) - 1, df)
X06_glmnet <- model.matrix(y ~ splines::ns(x1, 12) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)) - 1, df)

5b)

By default glmnet uses the lasso penalty. Fit a Lasso model by calling glmnet(). The first argument to glmnet() is the design matrix and the second argument is a regular vector for the response.

Train a Lasso model for the model 3 and model 6 formulations, assign the results to lasso_03 and lasso_06, respectively.

SOLUTION

lasso_03 <- glmnet(X03_glmnet, df$y)

lasso_06 <- glmnet(X06_glmnet, df$y)

5c)

Plot the coefficient path for each Lasso model by calling the plot() function on the glmnet model object. Specify the xvar argument to be 'lambda' in the plot() call.

SOLUTION

plot(lasso_03, xvar = 'lambda')

plot(lasso_06, xvar = 'lambda')

5d)

Now that you have visualized the coefficient path, it’s time to identify the best 'lambda' value to use! The cv.glmnet() function will by default use 10-fold cross-validation to tune 'lambda'. The first argument to cv.glmnet() is the design matrix and the second argument is the regular vector for the response.

Tune the Lasso regularization strength with cross-validation using the cv.glmnet() function for each model formulation. Assign the model 3 result to lasso_03_cv_tune and assign the model 6 result to lasso_06_cv_tune. Also specify the alpha argument to be 1 to make sure the Lasso penalty is applied in the cv.glmnet() call.

HINT: The random seed was assigned for you in two separate code chunks below. This will help ensure you can reproduce the cross-validation results.

SOLUTION

### the random seed is set for you
set.seed(812312)
### tune the model 3 formulation
lasso_03_cv_tune <- cv.glmnet(X03_glmnet, df$y, alpha = 1, nfolds = 10)

### the random seed is set for you
set.seed(812312)
### tune the model 6 formulation
lasso_06_cv_tune <- cv.glmnet(X06_glmnet, df$y, alpha = 1, nfolds = 10)

5e)

Plot the cross-validation results using the default plot method for each cross-validation result. How many coefficients are remaining after tuning?

SOLUTION

plot(lasso_03_cv_tune)

lasso_03_coef <- coef(lasso_03_cv_tune, s = "lambda.min")
num_coef_03 <- sum(lasso_03_coef != 0)-1
num_coef_03

## [1] 7

Apart from the intercept,there are 7 coefficients remaining after tuning in model 3.

plot(lasso_06_cv_tune)

lasso_06_coef <- coef(lasso_06_cv_tune, s = "lambda.min")
num_coef_06 <- sum(lasso_06_coef != 0)-1
num_coef_06

## [1] 2

Apart from the intercept，there are 2 coefficients remaining after tuning in model 6.

5f)

Which features have NOT been “turned off” by the Lasso penalty? Use the coef() function to display the lasso model cross-validation results to show the tuned penalized regression coefficients for each model.
Are the final tuned models different from each other?

SOLUTION

coef(lasso_03_cv_tune)

## 9 x 1 sparse Matrix of class "dgCMatrix"
##                         s1
## (Intercept)      0.3009999
## x1               .        
## I(x1^2)          .        
## x2               .        
## I(x2^2)         -0.3040403
## x1:x2            .        
## x1:I(x2^2)       .        
## I(x1^2):x2       .        
## I(x1^2):I(x2^2)  .

A dot indicates that the coefficient has been turned off. Apart from the intercept only the square of x2 remains.

coef(lasso_06_cv_tune)

## 65 x 1 sparse Matrix of class "dgCMatrix"
##                                       s1
## (Intercept)                    0.2818858
## splines::ns(x1, 12)1           .        
## splines::ns(x1, 12)2           .        
## splines::ns(x1, 12)3           .        
## splines::ns(x1, 12)4           .        
## splines::ns(x1, 12)5           .        
## splines::ns(x1, 12)6           .        
## splines::ns(x1, 12)7           .        
## splines::ns(x1, 12)8           .        
## splines::ns(x1, 12)9           .        
## splines::ns(x1, 12)10          .        
## splines::ns(x1, 12)11          .        
## splines::ns(x1, 12)12          .        
## x2                             .        
## I(x2^2)                       -0.2847331
## I(x2^3)                        .        
## I(x2^4)                        .        
## splines::ns(x1, 12)1:x2        .        
## splines::ns(x1, 12)2:x2        .        
## splines::ns(x1, 12)3:x2        .        
## splines::ns(x1, 12)4:x2        .        
## splines::ns(x1, 12)5:x2        .        
## splines::ns(x1, 12)6:x2        .        
## splines::ns(x1, 12)7:x2        .        
## splines::ns(x1, 12)8:x2        .        
## splines::ns(x1, 12)9:x2        .        
## splines::ns(x1, 12)10:x2       .        
## splines::ns(x1, 12)11:x2       .        
## splines::ns(x1, 12)12:x2       .        
## splines::ns(x1, 12)1:I(x2^2)   .        
## splines::ns(x1, 12)2:I(x2^2)   .        
## splines::ns(x1, 12)3:I(x2^2)   .        
## splines::ns(x1, 12)4:I(x2^2)   .        
## splines::ns(x1, 12)5:I(x2^2)   .        
## splines::ns(x1, 12)6:I(x2^2)   .        
## splines::ns(x1, 12)7:I(x2^2)   .        
## splines::ns(x1, 12)8:I(x2^2)   .        
## splines::ns(x1, 12)9:I(x2^2)   .        
## splines::ns(x1, 12)10:I(x2^2)  .        
## splines::ns(x1, 12)11:I(x2^2)  .        
## splines::ns(x1, 12)12:I(x2^2)  .        
## splines::ns(x1, 12)1:I(x2^3)   .        
## splines::ns(x1, 12)2:I(x2^3)   .        
## splines::ns(x1, 12)3:I(x2^3)   .        
## splines::ns(x1, 12)4:I(x2^3)   .        
## splines::ns(x1, 12)5:I(x2^3)   .        
## splines::ns(x1, 12)6:I(x2^3)   .        
## splines::ns(x1, 12)7:I(x2^3)   .        
## splines::ns(x1, 12)8:I(x2^3)   .        
## splines::ns(x1, 12)9:I(x2^3)   .        
## splines::ns(x1, 12)10:I(x2^3)  .        
## splines::ns(x1, 12)11:I(x2^3)  .        
## splines::ns(x1, 12)12:I(x2^3)  .        
## splines::ns(x1, 12)1:I(x2^4)   .        
## splines::ns(x1, 12)2:I(x2^4)   .        
## splines::ns(x1, 12)3:I(x2^4)   .        
## splines::ns(x1, 12)4:I(x2^4)   .        
## splines::ns(x1, 12)5:I(x2^4)   .        
## splines::ns(x1, 12)6:I(x2^4)   .        
## splines::ns(x1, 12)7:I(x2^4)   .        
## splines::ns(x1, 12)8:I(x2^4)   .        
## splines::ns(x1, 12)9:I(x2^4)   .        
## splines::ns(x1, 12)10:I(x2^4)  .        
## splines::ns(x1, 12)11:I(x2^4)  .        
## splines::ns(x1, 12)12:I(x2^4)  .

A dot indicates that the coefficient has been turned off. Apart from the intercept，only the square of x2 remains. So the final tuned models are the same.